Marked length rigidity for Fuchsian buildings
David Constantine
Jean-Fran\c{c
Izhar Oppenheim
We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT(-1) metrics on $X$ which are piecewise hyperbolic, and satisfy a natural non-singularity condition at vertices are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.
1503.01321.tex
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\title{Marked length rigidity for Fuchsian buildings}


\author{David Constantine}
\address{
Wesleyan University \\
Mathematics and Computer Science Department \\
Middletown, CT 06459}


\author{Jean-Fran\c{c}ois Lafont}
\address{Department of Mathematics\\
                 Ohio State University\\
                 Columbus, Ohio 43210}


%\author{Izhar Oppenheim}
%\address{Department of Mathematics\\
%                Ohio State University\\
%                 Columbus, Ohio 43210}





%\thanks{}



\date{\today}









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\begin{document}


\begin{abstract}

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial
automorphism group, which we assume act freely and cocompactly. We show that locally CAT(-1) metrics on $X$ which are piecewise
hyperbolic, and satisfy a natural non-singularity condition at vertices are marked length spectrum rigid within certain classes
of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.
\end{abstract}

\maketitle





\setcounter{secnumdepth}{1}



\setcounter{section}{0}

\section{\bf Introduction}

One of the central results in hyperbolic geometry is Mostow's rigidity theorem, which states that for closed hyperbolic 
manifolds of dimension $\geq 3$, isomorphism of fundamental groups implies isometry. Moving away from the constant
curvature case, one must impose some additional constraints on the isomorphism of fundamental groups
if one hopes to conclude it is realized by an isometry. On any closed negatively curved manifold $M$, each free
homotopy class of loops contains a unique geodesic representative. This gives a well-defined class function
$MLS: \pi_1(M)\rightarrow \mathbb R^+$, called the {\it marked length spectrum function}. Given a pair of negatively curved 
manifolds $M_0, M_1$, we say they have {\it the same} marked length spectrum if there is an isomorphism $\phi: \pi_1(M_0)
\rightarrow \pi_1(M_1)$ with the property that $MLS_1\circ \phi = MLS_0$. The marked length spectrum conjecture predicts
that closed negatively curved manifolds with the same marked length spectrum must be isometric (and that the 
isomorphism of fundamental groups is induced by an isometry). In full generality, the conjecture is only known to hold 
for closed surfaces, which was independently established by Croke \cite{croke} and Otal \cite{otal}. In the 
special case where one of the Riemannian metrics is locally symmetric, the conjecture was established by 
Hamenst\"adt \cite{ham} (see also Dal'bo and Kim \cite{dalbo-kim} for analogous results in the higher rank case).  

Of course, it is possible to formulate the marked length spectrum conjecture for other classes of geodesic 
spaces -- for example, compact locally CAT(-1) spaces. Still in the realm of surfaces, 
Hersonsky and Paulin \cite{hersonsky-paulin} extended the result  to some singular metrics on surfaces, while
Bankovi\'c and Leininger \cite{bankovic-leininger} and Constantine \cite{constantine} give extensions to the case of
non-positively curved metrics. Moving away from the surface case, the conjecture was verified independently by Alperin
and Bass \cite{alperin-bass} and by Culler and Morgan \cite{culler-morgan} in the special case of locally CAT(-1)
spaces whose universal covers are metric trees. This was recently extended by the authors to the context of
compact geodesic spaces of topological (Lebesgue) dimension one, see \cite{CL}. 

In this paper, we are interested in the marked length spectrum conjecture for compact quotients of 
{\it Fuchsian buildings}, a class of polygonal $2$-complexes supporting locally CAT(-1) metrics. Fixing such 
a quotient $X$, we can then look at various families of locally negatively curved metrics on $X$. The 
metrics we consider are {\it piecewise Riemannian}: each polygon in the complex is 
equipped with a Riemannian metric with geodesic boundary edges. They are also assumed to be locally negatively curved,
which means that the metrics satisfy Gromov's ``large link condition'' at all the vertices. We consider three classes of such
metrics: those whose curvatures are everywhere bounded above by -1, those whose curvature is everywhere hyperbolic,
and those whose curvatures are everywhere within the interval $[-1, 0)$. The space of such metrics will be denoted
$\mathcal M_{\leq} (X)$, $\mathcal M_{\equiv}(X)$, and $\mathcal M_{\geq}(X)$ respectively. Note that the family of piecewise
hyperbolic metrics $\mathcal M_{\equiv}(X)$ are precisely the metrics lying in the intersection $\mathcal M_{\leq}(X) \cap
\mathcal M_{\geq}(X)$. Furthermore, all three of these classes
of metrics lie within the space $\mathcal M_{neg}(X)$, consisting of all (locally) negatively curved, piecewise Riemannian
metrics on $X$.
Finally, if we impose some further regularity conditions on the vertices, we obtain subclasses of metrics $\mathcal M_{\leq}^v(X)$, $\mathcal M_{\equiv}^v(X)$, $\mathcal M_{\geq}^v(X)$, and $\mathcal M_{neg}^v(X)$. 
We refer our reader to Section \ref{background} for 
further background on Fuchsian buildings, including precise definitions for these classes of metrics -- let us just mention
that, amongst these, the most ``regular'' metrics are those lying in the class $\mathcal M_{\equiv}^v(X)$, 
which forms an analogue of Teichm\"uller space for $X$.

\vskip 10pt

\begin{theorem}
Let $X$ be a quotient of a Fuchsian building $\tilde X$ by a subgroup $\Gamma \leq \aut (\tilde X)$ of the combinatorial
automorphism group $\aut (\tilde X)$ which acts
freely and cocompactly.
Consider a pair of negatively curved metrics $g_0, g_1$ on $X$, where $g_0$ is in $\mathcal M_{\equiv}^v(X)$, 
and $g_1$ is in $\mathcal M_{\geq}^v(X)$.  Then $(X, g_0)$ and $(X, g_1)$ have the 
same marked length spectrum if and only if they are isometric.
\end{theorem}

In the process of establishing the {\bf Main Theorem}, we also obtain a number of auxiliary results which may be of some
independent interest. Let us briefly mention a few of these. Throughout the rest of this section, $X$ will denote a quotient 
of a Fuchsian building $\tilde X$ by a subgroup $\Gamma \leq \aut (\tilde X)$ which acts
freely and cocompactly.

The first step is to obtain marked length spectrum rigidity for
pairs of metrics in $\mathcal M_{\leq}^v(X)$.

\begin{thm}[MLS rigidity -- special case]\label{thm:MLS-rigidity-special-case}
Let $g_0 , g_1$ be any two metrics in $\mathcal M_\equiv ^v$ and $\mathcal M_{\leq}(X)$ respectively. 
Then $(X, g_0)$ and $(X, g_1)$ have the same marked length spectrum if and only if they are isometric.
\end{thm}

This result is established in Section 3, and is based on an argument outlined to us by an anonymous referee.  
Next, we study the volume functional on the space of metrics. We note that the volume is constant on 
the subspace $\mathcal M_{\equiv}^v(X)$, and in Section 4, we show the following rigidity result:

\begin{thm}[Minimizing the volume]\label{thm:minimizing-the-volume}
Let $g_0$ be a metric in $\mathcal M_{\equiv}^v(X)$, and $g_1$ an arbitrary metric in $\mathcal M_{\geq}^v(X)$.
If $Vol(X, g_1) \leq Vol(X, g_0)$, then $g_1$ must lie within $\mathcal M_{\equiv}^v(X)$ (and the inequality
is actually an equality).
\end{thm}

Finally, the last (and hardest) step in the proof is a general result relating the marked length spectrum
and the volume. We show:

\begin{thm}[MLS determines volume]\label{thm:MLS-determines-volume}
Let $g_0, g_1$ be an arbitrary pair of metrics in $\mathcal M_{neg}^v(X)$. 
If $MLS_0 \leq MLS_1$, then $Vol(X, g_0) \leq Vol(X, g_1)$.
\end{thm}

The analogous result for negatively curved metrics on a closed surface is due to Croke and Dairbekov 
\cite{croke-dairbekov}, who also established a version for conformal metrics on negatively curved manifolds (see 
also some related work by Fana\"i \cite{Fa} and by Z. Sun \cite{sun}). Our proof of Theorem \ref{thm:MLS-determines-volume} roughly follows the approach in \cite{croke-dairbekov} -- after setting up the preliminaries in Sections \ref{sec:pairing} and \ref{sec:cont},  we prove the theorem in Section \ref{vol_conject}.


Finally, using these three theorems, the proof of the {\bf Main Theorem} is now straightforward.

\begin{proof}[Proof of Main Theorem]
Let $g_0$ be a metric in $\mathcal M_{\equiv}^v(X)$, and $g_1$ a metric in $\mathcal M^v_{\geq}(X)$. If $MLS_0 \equiv MLS_1$,
then by Theorem \ref{thm:MLS-determines-volume}, we see that $Vol(g_1)=Vol(g_0)$. 
So Theorem \ref{thm:minimizing-the-volume} forces $g_1$ to lie in the space $\mathcal M_{\equiv}^v(X)$. 
Since they have the same marked length spectrum, Theorem \ref{thm:MLS-rigidity-special-case} 
now allows us to conclude that $(X, g_0)$ is isometric to $(X, g_1)$, completing the proof.
\end{proof}

These results provide partial evidence towards the general marked length spectrum conjecture for these compact quotients of Fuchsian buildings, which we expect to hold for any pairs of metrics in $\mathcal M_{neg}(X)$. We should mention that rigidity theorems for such quotients $X$ are often difficult to prove. For instance combinatorial (Mostow) rigidity was established by Xiangdong Xie \cite{X} (building on previous work of Bourdon \cite{bourdon-3}). Quasi-isometric rigidity was also established by Xie \cite{X}, generalizing earlier work of Bourdon and Pajot \cite{BP}. Superrigidity with targets in the isometry group of $\tilde X$ was established by Daskalopoulos, Mese, and Vdovina \cite{DMV}. Finally, in the context of volume entropy, recent work of Ledrappier and Lim \cite{LL} leave us uncertain as to which metrics in $\mathcal M_{\equiv}(X)$ minimize the volume growth entropy (they show that the ``obvious'' candidate for a minimizer is actually not a minimizer).



\subsection{Acknowledgements} The first named author would like to thank Ohio State for hosting him for a visit during which a portion of this work was completed. The second author was partially supported by the NSF, under grant DMS-1207782.
The authors would also like to thank the anonymous referee for informing us of Bourdon's work, and suggesting the proof
of Theorem \ref{thm:MLS-rigidity-special-case} which is given in Section 3. Our original argument for this result was 
considerably more involved, and also included some unnecessary assumptions. The authors are also
indebted to the referee for asking probing questions, which led the authors to discover a serious gap in an earlier
draft of the paper. Finally, we would also like to thank Marc Bourdon and Alina Vdovina for helpful comments.






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\section{Background material}\label{background}


\subsection{Fuchsian buildings.}
We start by summarizing basic notation and conventions on Fuchsian buildings, which were first introduced by Bourdon
\cite{Bou}. These are $2$-dimensional polyhedral complexes which satisfy a number of axioms. First, one starts with a 
compact convex hyperbolic polygon $R\subset \mathbb H^2$, with each angle of the form $\pi/m_i$ for some $m_i$
associated to the vertex ($m_i\in \mathbb N,
m_i\geq 2$). Reflection in the geodesics extending the sides of $R$ generate a Coxeter group $W$, and the orbit of $R$ under
$W$ gives a tessellation of $\mathbb H^2$. Cyclically labeling the edges of $R$ by the integers $\{1\}, \ldots, \{k\}$ (so that the
vertex between the edges labelled $i$ and $i+1$ has angle $\pi/m_i$), one can apply the $W$ action to obtain a $W$-invariant labeling
of the tessellation of $\mathbb H^2$; this edge labeled polyhedral $2$-complex will be denoted $A_R$, and called the {\it model 
apartment}.

A polygonal $2$-complex $\tilde X$ is called a $2$-dimensional hyperbolic building if it contains an edge labeling by the integers
$\{1, \ldots , k\}$, along with a distinguished collection of subcomplexes $\mathcal A$ called the {\it apartments}. The individual
polygons in $\tilde X$ will be called {\it chambers}. The complex is required to have the following properties:
\begin{itemize}
\item each apartment $A\in \mathcal A$ is isomorphic, as an edge labeled polygonal complex, to the model apartment $A_R$,
\item given any two chambers in $\tilde X$, one can find an apartment $A\in \mathcal A$ which contains the two chambers, and
\item given any two apartments $A_1, A_2\in \mathcal A$ that share a chamber, there is an isomorphism of labeled 
$2$-complexes $\varphi: A_1\rightarrow A_2$ that fixes $A_1\cap A_2$. 
\end{itemize}
If in addition each edge labeled $i$ has a fixed number $q_i$ of incident polygons, then $\tilde X$ is called a {\it Fuchsian building}.
The group $\aut(\tilde X)$ will denote the group of combinatorial (label-preserving) automorphisms of the Fuchsian building $\tilde X$. 


Throughout this paper we make the standing assumption that $\tilde X$ is \emph{thick}, i.e. that every edge is contained in at least three chambers. Thus, the overall geometry of the building $\tilde X$ will involve an interplay between the geometry of the apartments, and
the combinatorics of the branching along the edges.

Note that making each polygon in $\tilde X$ isometric to $R$ via the label-preserving map produces a CAT(-1) metric on $\tilde X$.
However, a given polygonal $2$-complex might have several metrizations as a Fuchsian building: these correspond to 
varying the hyperbolic metric on $R$ while preserving the angles at the vertices. Any such variation induces a new 
CAT(-1) metric on $\tilde X$. The hyperbolic polygon $R$ is called {\it normal} if it has an inscribed circle that touches all its sides --
fixing the angles of a polygon to be $\{\pi/m_1, \ldots ,\pi/m_k\}$, there is a unique normal hyperbolic polygon with those 
given vertex angles.  We will call the quantity $\pi/m_i$ the {\it combinatorial angle} associated to the corresponding vertex.
A Fuchsian building will be called {\it normal} if all metric angles are equal to the corresponding combinatorial angles {\it and} 
the metric on each chamber is normal.
We can now state Xiangdong Xie's version of Mostow rigidity for Fuchsian buildings (see \cite{X}):

\begin{thm}[Xie]\label{xie}
Let $\tilde X_1, \tilde X_2$ be a pair of Fuchsian buildings, and let $\Gamma_i\leq \Isom(\tilde X_i)$ be a uniform lattice.
Assume that we have an isomorphism $\phi: \Gamma_1\rightarrow 
\Gamma_2$. Then there is a $\phi$-equivariant homeomorphism $\Phi: \tilde X_1\rightarrow \tilde X_2$.
Moreover, if both buildings are normal, then one can choose $\Phi$ to be a $\phi$-equivariant isometry.
\end{thm}

Another notion that will reveal itself useful is the following: inside $\tilde X$, we have a collection of {\it walls}, which are defined
as follows. First, recall that each apartment in the building is (combinatorially) modeled on a $W$-invariant 
polygonal tessellation of $\mathbb H^2$. The geodesics extending the various 
sides of the polygons give a $W$-invariant collection of geodesics in $\mathbb H^2$, which are also a collection of 
combinatorial paths in the tessellation. This gives a distinguished collection of combinatorial paths in the model apartment $A_R$. 
Via the identification of apartments $A \in \mathcal A$ in $\tilde X$ with the model apartment $A_R$, we obtain the notion of wall in an
apartment of $\tilde X$. Note that every edge in $\tilde X$ is contained in many different walls of $\tilde X$.

\subsection{Structure of vertex links}
For a Fuchsian building, the combinatorial axioms force some additional structure on the links of vertices: these graphs 
must be (thick) {\it generalized $m$-gons}  (see for instance \cite[Prop. 4.9 and 4.44]{brown}). Work of Feit and Higman \cite{FH} then implies that each $m_i$ must 
lie in the set $\{2, 3, 4, 6, 8\}$. Viewed as a combinatorial graph, a generalized $m$-gon has diameter $m$ and girth $2m$. Moreover, taking the 
collection of cycles of length $2m$ within the graph to be the set of apartments, such a graph has the structure of a (thick) 
spherical building (based on the action of the dihedral group $D_{2m}$ of order $2m$ acting on $S^1$). 

For instance, when $m=2$, a generalized $2$-gon is just a complete bipartite graph $K_{p,q}$. When $m=3$, generalized $3$-gons
correspond to the incidence structure on finite projective planes (whose classification is a notorious open problem). When $m>3$, examples
are harder to find. An extensive discussion of generalized $4$-gons can be found in the book \cite{payne-thas}. For generalized $6$-gons
and $8$-gons, the only known examples arise from certain incidence structures associated to some of the finite groups of Lie 
type (see e.g. \cite{van-Maldeghem}).

Note that, at a given vertex $v$, the edges incident to $v$ always have one of two possible (consecutive) labels. 
On the level of the link, this means that $lk(v)$ comes equipped with an induced $2$-coloring of the vertices by the integers $i, i+1$. 
Since all edges with a given label 
$i$ have $q_i$ incident chambers, this means that the vertices in $lk(v)$ colored $i,i+1$ have degrees $q_i, q_{i+1}$ respectively. In the case
of generalized $2$-gons, the vertex $2$-coloring is the one defining the complete bipartite graph structure. For a generalized $3$-gon, 
the identification of the graph with the incidence structure of a finite projective plane $\mathcal P$ provides the $2$-coloring: the colors 
determine whether a vertex in the graph corresponds to a point or to a line in $\mathcal P$.

Split the vertex set into $\mathcal V_i, \mathcal V_{i+1}$, the set of vertices with label $i, i+1$ respectively. From the bipartite 
nature of the graph, the number of edges in the graph satisfies $|\mathcal E| = q_i |\mathcal V_i| = q_{i+1}|\mathcal V_{i+1}|$. 
Given an edge $e\in \mathcal E$, we now count the number of apartments
(i.e. $2m$-cycles) passing through $e$. In a generalized $m$-gon, any path of length $m+1$ is contained in a unique apartment 
(see, e.g. \cite[Prop. 7.13]{Weiss-structure}). Thus, to count the number of apartments through $e$, it is enough to count the number of ways to
extend $e$ to a path of length $m+1$. The number of branches we can take at each vertex alternates between a $q_i$ and a 
$q_{i+1}$. So if $m$ is even, we obtain that the number of edges is $N:= q_i^{m/2}q_{i+1}^{m/2}$. 

If $m=3$ is odd, then we note
that $q_i = q_{i+1}$. Indeed, opposite vertices in one of the $6$-cycles have labels $q_i$ and $q_{i+1}$. But for each vertex in $lk(v)$ 
(which corresponds to an edge in the original building) the valence corresponds to the number of chambers which 
share that edge. Since the branching in the ambient building occurs along walls, for two opposite vertices in an apartment 
in the link, the valence must be the same. So in this case, the number of apartments through an edge is $N:= q_i^3 = q_{i+1}^3$.

%Note that, the number of apartments through an edge is {\it independent} of the choice of edge.
%If $\mathcal A$ denotes the apartments (i.e. $2m$-cycles) in the 
%graph, and $N$ denotes the number of apartments through an edge, then we also have that $|E|=m|\mathcal A|/N$.


\subsection{Spaces of metrics}

Now consider a compact quotient $ X = \tilde X/\Gamma$ of a Fuchsian building, where $\Gamma\subset Aut(\tilde X)$ is a lattice in the
group of combinatorial automorphisms of $\tilde X$. On the quotient space $X$, we will consider metrics which are {\it piecewise
Riemannian}, i.e. whose restriction to each chamber of $X$ is a Riemannian metric, such that all the sides of
the chamber are geodesics. Moreover, we will restrict to metrics which are locally negatively curved -- and thus will require the
metrics on each chamber to have sectional curvature $<0$. We will denote this class of metrics by 
$\mathcal{M} _{neg}$. If we instead require each chamber to be hyperbolic (i.e. to have curvature $\equiv -1$), 
then we obtain the space $\mathcal{M}_{\equiv}$. Similarly, we can require each chamber to have curvature $\leq -1$,
or curvature in the interval $[-1, 0)$. These give rise to the corresponding spaces $\mathcal{M}_{\leq}$ or $\mathcal{M}_{\geq}$
respectively. Clearly, we have a proper inclusion $\mathcal M_\leq \cup \mathcal{M}_{\equiv} \cup \mathcal M_\geq \subset
\mathcal{M}_{neg}$, as well as the equality $\mathcal{M}_\equiv = \mathcal M_\leq \cap \mathcal M_\geq$. Notice that, for 
all of these classes of metrics, the negative curvature property imposes some constraints on the metric near the vertices
of $X$: they must always satisfy Gromov's ``large link condition" (see discussion below).

In order to obtain a true analogue of hyperbolic metrics on $X$, one needs to impose some additional regularity condition. 
To illustrate this, consider the case of piecewise hyperbolic metrics on ordinary surfaces. One can pullback 
a hyperbolic metric on a surface $\Sigma _2$ of genus two via a degree two map $\Sigma_4\rightarrow \Sigma_2$ 
ramified over a pair of points. The resulting metric on the surface $\Sigma_4$ of genus four is piecewise hyperbolic, 
but has two singular points with cone angle $=4\pi$, so in particular is {\it not} hyperbolic.
By analogy, an analogue of a constant curvature metric on $X$ 
should have ``as few'' singular points as possible. 

Of course, the only possible singularities occur at the vertices
of $X$. Given a vertex $\tilde v \in \tilde X$, one has several 
apartments passing through $\tilde v$, and one can restrict the metric to each of these apartments. The negative curvature condition implies that each of these apartments inherits a (possibly singular) negatively curved metric. This tells us that the sum of the angles around the vertex $\tilde v$ in each apartment is $\geq 2\pi$. We say that the vertex $\tilde v$ is {\it metrically
non-singular} if, when restricted to each apartment through $\tilde v$, the sum of the angles at $\tilde v$ is exactly $2\pi$. 
A metric has {\it non-singular vertices} if every vertex is
metrically non-singular. We will denote the subspace of such metrics inside
$\mathcal{M}_{neg}$ by $\mathcal{M}_{neg}^{v}$. We can similarly define the subsets $\mathcal{M}_{\leq}^v$, $\mathcal M_\equiv^v$ and $\mathcal M_\geq^v$ inside the spaces $\mathcal{M}_{\leq}, \mathcal M_\equiv , \mathcal M_\geq$ respectively
(the superscript $v$ is intended to denote non-singular vertices).

When $X$ is equipped with a piecewise Riemannian metric $g$, each vertex link $lk(v)$ gets an induced metric $d$.
Indeed, an edge in $lk(v)$ corresponds to a chamber corner in $X$. Since the chamber $C$ has a Riemannian metric with geodesic sides, the corner has an angle $\theta$ measured in the $g$ metric. The $d$-length of the corresponding edge is defined to be the angle $\theta$. With respect to this metric, the negative curvature condition at $v$ translates to saying that every $2m$-cycle in the generalized $m$-gon $lk(v)$ has total $d$-length $\geq 2\pi$ (Gromov's ``large link'' condition).  The metric $g\in \mathcal{M}_{neg}$ lies in the subclass $ \mathcal{M}_{neg}^{v}$ precisely if for every vertex link $lk(v)$, the metric $d$-length of {\it every} $2m$-cycle is {\it exactly} $2\pi$. Of course, a similar statement holds for 
$\mathcal{M}_{\leq}^{v}, \mathcal M^v_\equiv , \mathcal M^v _\geq$. 
As we will see below (Corollary \ref{metric-angles-equal-comb-angles}), the non-singularity condition on vertices imposes very strong constraints on the vertex angles -- they will always equal the corresponding combinatorial angle.


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\section{MLS rigidity for metrics in $\mathcal{M}_{\equiv}^{v}$ } 

This section is devoted to proving Theorem \ref{thm:MLS-rigidity-special-case}. The argument we present here was suggested
to us by the anonymous referee. We start by reminding the reader of some metric properties of boundaries of CAT(-1) spaces. If $(\tilde X,d)$ is any CAT(-1) space, with boundary at infinity $\partial ^\infty (\tilde X,d)$, the {\it cross-ratio} is a function on $4$-tuples $(\xi, \xi', \eta, \eta')$ of distinct points in $\partial ^\infty (\tilde X,d)$. It is defined by:
$$[\xi \xi' \eta \eta'] := \lim_{(a, a', b, b') \to (\xi, \xi', \eta, \eta')} \text{Exp}\left(\frac{1}{2}\left(d(a, b) + d(a', b') - d(a, b')-d(a',b) \right)\right)$$
and the $4$-tuple $(a, a', b, b')$ converges radially towards $(\xi, \xi', \eta, \eta')$. If $\tilde Y$ is another CAT(-1) space, a topological embedding $\Phi: \partial ^\infty \tilde Y \rightarrow \partial ^\infty \tilde X$ is a {\it M\"obius map} if it respects the cross-ratio, i.e. for all $4$-tuples of distinct points $(\xi, \xi', \eta, \eta')$ in $\partial ^\infty Y$, we have
$$[\Phi(\xi) \Phi(\xi') \Phi(\eta) \Phi(\eta')] = [\xi \xi' \eta \eta'].$$
Note that an isometric embedding of CAT(-1) spaces automatically induces a M\"obius map between boundaries at infinity. As a consequence, for a totally geodesic subspace of a CAT(-1) space, the intrinsic cross-ratio (defined from within the subspace) coincides with the extrinsic cross-ratio (restriction of the cross-ratio from the ambient space).



\begin{proof}[Proof of Theorem \ref{thm:MLS-rigidity-special-case}]
Lifting the metrics $g_0, g_1$ to the universal cover, the identity map lifts to a quasi-isometry $\Phi: (\tilde X, \tilde g_0) \rightarrow (\tilde X, \tilde g_1)$. This induces a map between boundaries at infinity $\partial ^\infty \Phi: \partial ^\infty (\tilde X, \tilde g_0) \rightarrow \partial ^\infty (\tilde X, \tilde g_1)$. Otal showed that, if an isomorphism of fundamental groups preserves the marked length spectrum, then the induced map on the boundaries at infinity is M\"obius (see \cite{otal-2} -- the argument presented there is for negatively curved closed manifolds, but the proof extends verbatim to the $CAT(-1)$ setting).


Now let $\mathcal A$ be the collection of apartments in the building $\tilde X$ (note that this is independent of the choice of metric on $X$). Since $g_0\in \mathcal M_\equiv ^v(X)$, each apartment $A \subset \tilde X$ inherits a piecewise hyperbolic metric, with no singular vertices. So each $(A, \tilde g_0|_A)$ is a totally geodesic subspace of $(\tilde X, \tilde g_0)$, isometric to $\mathbb H^2$. The map $\partial ^\infty \Phi$ sends the circle corresponding to $\partial ^\infty (A, \tilde g_0|_A)$ to the circle in $\partial ^\infty (\tilde X, \tilde g_1)$ corresponding to the totally geodesic subspace  $(A, \tilde g_1|_A)$ (see \cite{X}). Since the map $\partial ^\infty \Phi$ preserves the cross-ratio, work of Bourdon \cite{bourdon} implies that there is an isometric embedding $F_A:(A, \tilde g_0|_A) \rightarrow (\tilde X, \tilde g_1)$ which ``fills-in'' the boundary map. This isometry must have image $(A, \tilde g_1|_A)$, which hence must also be isometric to $\mathbb H^2$. Applying this to every apartment, we see that the metric $g_1$, which was originally assumed to be in $\mathcal M_{\leq}(X)$, must actually lie in the subspace $\mathcal M^v_{\equiv}(X)$.

Finally, we claim that there is an equivariant isometry between $(\tilde X, \tilde g_0)$ and $(\tilde X, \tilde g_1)$. For each apartment $A\in \mathcal A$, we have an isometry $F_A:(A, \tilde g_0|_A) \rightarrow (A, \tilde g_1|_A)$. From Xiangdong Xie's work, the 
boundary map $\partial ^\infty F_A \equiv \partial ^\infty \Phi |_{\partial ^\infty A}$ maps endpoints of walls to endpoints of
walls (see \cite[Lemma 3.11]{X}), so the isometry $F_A$ respects the tessalation of the apartment $A$, i.e. sends chambers in $A$ isometrically onto chambers in $A$. But a priori, we might have two different apartments $A, A'$ with the property that $F_A$ and $F_{A'}$ send a given chamber to two distinct chambers. So in order to build a global isometry from $(\tilde X, \tilde g_0)$ to $(\tilde X, \tilde g_1)$, we still need to check that the collection of maps $\{F_A\}_{A\in \mathcal A}$ are compatible. 

Given any two apartments $A, A' \in \mathcal A$ with non-empty intersection $A \cap A' = K$, we want to check that the maps $F_A$ and $F_{A'}$ coincide on the set $K$. Let us first consider the case where $K$ is a half-space, i.e. there is a single wall $\gamma$ lying in $A\cap A'$, and $K$ coincides with the subset of $A$ (respectively $A'$) lying to one side of $\gamma$. In this special case, it is easy to verify that $F_A$ and $F_{A'}$ restrict to the same map on $K$. Indeed, Bourdon constructs the map $F_A$ as follows: given a point $p\in K$ take any two geodesics $\eta, \xi$ passing through $p\in (A, \tilde g_0)$, look at the corresponding pair of geodesics $\eta', \xi'$ in $(A, \tilde g_1)$ (obtained via the boundary map), and define $F_A(p):= \eta'\cap \xi'$. Bourdon argues that this intersection is non-empty, and independent of the choice of pairs of geodesics. The map $F_{A'}$ is defined similarly. But now if $p\in \text{Int}(K)$, one can choose a pair of geodesics $\eta, \xi \subset \text{Int}(K)$. Since $\text{Int}(K)$ is contained in both $A, A'$, this pair of geodesics can be used to see that $F_A(p)= \eta' \cap \xi' = F_{A'}(p)$. This shows that, if $K= A\cap A'$ is a half-space, then $F_A|_K \equiv F_{A'}|_K$. 

For the general case, we now assume that we have a pair of apartments $A, A'$ with the property that $A\cap A'=K$ contains a chamber, and let $x$ be an interior point of this chamber. Then work of Hersonsky and Paulin \cite[Lemma 2.10]{hersonsky-paulin} produces a sequence of apartments $\{A_i\}_{i\in \mathbb N}$ with the property that $A_1=A$, each $A_i\cap A_{i+1}$ is a half-space containing $x$, and the $A_i$ converge to $A'$ in the topology of uniform convergence on compacts. From the discussion in the previous paragraph, one concludes that $F_A(x)= F_{A_i}(x)$ for all $i\in \mathbb N$, and from the uniform convergence, it is easy to deduce that $F_{A'(x)} = \lim F_{A_i}(x) = F_A(x)$. This verifies that the maps $\{F_A\}_{A\in \mathcal A}$ all coincide on a full-measure set (the interior points to chambers), and hence patch together to give a global isometry $F: (\tilde X, \tilde g_0)\rightarrow (\tilde X, \tilde g_1)$. Equivariance of the isometry follows easily from the naturality of the construction, along with the geometric nature of the maps $F_A$. Descending to the compact quotient completes the proof of Theorem \ref{thm:MLS-rigidity-special-case}.
\end{proof}

\begin{rem}
The argument presented here relies crucially on Bourdon's result in \cite{bourdon}. In the proof of the latter, the normalization of the spaces under consideration is important. The hyperbolic space mapping in must have curvature which matches the upper bound on the curvature in the target space. This is the key reason why the argument presented here does not immediately work in the setting of the {\bf Main Theorem}, where the metric $g_1$ is assumed to have piecewise curvature $\geq -1$.
\end{rem}


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\section{$\mathcal M_{\equiv}^v(X)$ minimizes the volume}

This section is dedicated to proving Theorem \ref{thm:minimizing-the-volume}.
For a vertex $v$ in our building, let $lk(v)$ denote the link of the vertex.  Combinatorially, this link is a  
generalized $m$-gon, hence a $1$-dimensional spherical building.  The edges of the generalized $m$-gon 
correspond to the chamber angles at $v$, and so any piecewise Riemannian metric on the building induces a metric on the link:
\[ d_i: E(lk(v)) \to \mathbb{R}^+ .\nonumber\]
For these metrics, $Vol( -,d_i)$ is simply the sum of all edge lengths. We first argue that the vertex regularity hypothesis
strongly constrains the angles.

\begin{lem}\label{combinatorial-angles}
Let $\mathcal G$ be a thick generalized $m$-gon. Assume we have a metric $d$ on $\mathcal G$ with the property that 
every $2m$-cycle in $\mathcal G$ has length exactly $2\pi$. Then every edge has length $\pi/m$.
\end{lem}

\begin{proof}
Consider a pair of vertices $v, w$ in $\mathcal G$ at combinatorial distance $=m$. Let $\mathcal P$ denote the set of all paths of combinatorial
length $m$ joining $v$ to $w$. Note that, since any two paths in $\mathcal P$ have common endpoints at $v,w$, they cannot have any
other vertices in common -- for otherwise one would find a closed loop of length $<2m$, which is impossible. The concatenation of any
two paths in $\mathcal P$ form a $2m$-cycle, so has length exactly $2\pi$. By the thickness hypothesis, there are at least three such 
paths, hence every path in $\mathcal P$ has metric length $=\pi$. Applying this argument to all pairs of antipodal vertices in $\mathcal G$, we 
see that {\it every} path in $\mathcal G$ of combinatorial length $m$ has metric length $=\pi$.

Now let us return to our original pair $v,w$. Every edge emanating from $v$ can be extended to a (unique) 
combinatorial path of length $m$ terminating at $w$ (and likewise for edges emanating from $w$). This gives a bijection between
edges incident to $v$ and edges incident to $w$. Let $e_i^w$ denote the edge incident to $w$ associated to the edge $e_i^v$ incident
to $v$. Choosing $i\neq j$, we have a $2m$-cycle obtained by concatenating the paths $\mathbf{p}_i$ and $\mathbf{p}_j$ of combinatorial
length $m$, joining $v$ to $w$ and passing through $e_i^v, e_j^w$. Within this $2m$-cycle, we have a path of combinatorial length
$m-1$ which can be extended, at each endpoint, by $e_i^v, e_j^w$ respectively. Since every path of combinatorial length $m$ has 
metric length exactly $\pi$, we see that the edges $e_i^v, e_j^w$ must have the same metric length. By the thickness hypothesis, we
have $\deg(v)=\deg(w) \geq 3$, and it follows that every edge at the vertex $v$ has exactly the same metric length. 

Using the same argument at every vertex, and noting that $\mathcal G$ is a connected graph, we see that every edge in 
$\mathcal G$ has exactly the same metric length. Finally, from the fact that the $2m$-cycles have length $=2\pi$, we see that 
this common length must be $=\pi/m$.
\end{proof}

Applying Lemma \ref{combinatorial-angles} to the links of each vertex in $X$, gives us:

\begin{cor}\label{metric-angles-equal-comb-angles}
If $g\in \mathcal{M}_{neg}^{v}$, then at every vertex $v\in X$, all the metric angles are equal to the combinatorial angles.
\end{cor}

Recall that the area of a hyperbolic (geodesic) polygon, by the Gauss-Bonnet formula, is completely determined by the number of
sides and the angles at the vertices. So we also obtain:

\begin{cor}\label{constant-volume}
The volume functional is constant on the space $\mathcal{M}_{\equiv}^{v}(X)$.
\end{cor}
 
We are now ready to establish Theorem \ref{thm:minimizing-the-volume}


\begin{proof}[Proof of Theorem \ref{thm:minimizing-the-volume}] 
We will argue by contradiction. Assume we have a metric $g_1\in \mathcal M^v_\geq(X) \setminus \mathcal M^v_\equiv (X)$ 
with the property that $Vol(X, g_1)\leq Vol(X, g_0)$. 
Applying the Gauss-Bonnet theorem to any chamber $C$, we obtain for either metric that
\[ \int_C K_i dvol_i = -\pi(n-2) + \sum_{j=1}^n\theta^{(j)}_i\]
where $n$ is the number of sides for any chamber, $\theta_i^{(j)}$ are the interior angles of $C$, and $K_i$ is the curvature
function for the metric $g_i$.  Denote
by $\mathcal P(X)$ the collection of chambers in $X$. For the whole space $X$, we have
\begin{equation}\label{gauss_bonnet}
\sum_{C\in \mathcal P(X)} \int_C K_i dvol_i = -|\mathcal P(X)|\pi(n-2) + \sum_{C\in \mathcal P(X)} \sum_{j=1}^n\theta^{(j)}_i.
\end{equation}
Under the assumptions of the Theorem, we have
\begin{align*}
\sum_{C\in \mathcal P(X)} \int_C K_0 dvol_0 & = \sum_{C\in \mathcal P(X)} \int_C -1 \hskip 2pt dvol_0 \\
& = - Vol(X, g_0) \\
& \leq - Vol(X, g_1) \\
& =  \sum_{C\in \mathcal P(X)} \int_C -1 \hskip 2pt dvol_1 \\
& < \sum_{C\in \mathcal P(X)} \int_C K_1 dvol_1.
\end{align*}
(The last inequality is strict, since from the assumption that $g_1\in  \mathcal M^v_\geq(X) \setminus \mathcal M^v_\equiv (X)$, 
there must be at least one interior point on some chamber where the curvature $K_1$ is greater than $-1$.)
Since the quantity $-|\mathcal P(X)|\pi(n-2)$ is independent of the choice of metric, applying equation (\ref{gauss_bonnet}) gives us
\[ \sum_{C\in \mathcal P(X)} \sum_{j=1}^n\theta^{(j)}_0 <  \sum_{C\in \mathcal P(X)} \sum_{j=1}^n\theta^{(j)}_1. \]
But each of these two sums can be interpreted as $\sum_{v}Vol(lk(v), d_i)$ for the respective metrics.
Hence, there must be at least one vertex $v$ whose $d_0$-volume is strictly smaller than its $d_1$-volume.
But by Corollary \ref{metric-angles-equal-comb-angles}, the vertex regularity hypothesis forces the volumes of the links to be equal, a contradiction. This completes the proof of the Theorem \ref{thm:minimizing-the-volume}.
\end{proof}









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\section{Geodesic flows and geodesic currents on Fuchsian buildings}\label{sec:flows and currents}


In this section, we set up the terminology needed for the proof of Theorem 3.


\subsection{Geodesic flow}
Let $\tilde X$ be a hyperbolic building, equipped with a $CAT(-\epsilon^2)$ metric for some $\epsilon>0$, and $X= \tilde X / \Gamma$ a compact quotient of $\tilde X$ by 
a lattice $\Gamma \leq \aut(X)$ in the group of combinatorial automorphisms, acting freely, isometrically, and cocompactly. We make the following definitions:

\begin{itemize}
	\item Let $G(\tilde X)$ be the set of unit-speed parametrizations of geodesics in $\tilde X$.  Since $\tilde X$ is $CAT(-\epsilon^2)$, 
	$G(\tilde X) = (\partial^\infty \tilde X\times \partial^\infty \tilde X \times \mathbb{R})\setminus (\Delta\times 
	\mathbb R)$ where $\Delta$ is the  
	diagonal in $\partial^\infty \tilde X\times \partial^\infty \tilde X$. The quotient space
	$G(X):=G(\tilde X)/\Gamma$ by the naturally induced $\Gamma$-action is the space of unit-speed 
	geodesic parametrizations on $X= \tilde X/\Gamma$.  
	\item As in \cite[Section 3]{ball_brin}, let $S'$ denote the set of all unit length vectors based at a point in $X^{(1)}\setminus X^{(0)}$ (i.e. at an edge but not a vertex) and pointing 
	into a chamber.  $S'C$ is the set pointing into a particular chamber $C$.  $S'_xC$ is the set pointing into $C$ and based 
	at $x$.  $S'_x = \cup S'_xC_i$ is the union over all chambers adjacent to $x$.  %There is a slight difference here from 
	%\cite{ball_brin} in that we allow vectors based at vertices (which are excluded in Ballmann and Brin's definitions). We make this adjustment because closed geodesics which hit vertices, in particular paths along $X^{(1)}$, are the key to our proof of Theorem \ref{thm:MLS-rigidity-special-case}.
	\item  For $v\in S'C$, let $I(v)\in S'C$ to be the vector tangent to the geodesic segment through C generated by $v$ and pointing the 
	opposite direction.  Let $F(v) \subset S'$ be the set of all vectors based at the footpoint of $I(v)$ which geodesically extend 
	the segment defined by $v$.  Let $W$ be the set of all bi-infinite sequences $(w_n)_{n\in \mathbb{Z}}$ such that 
	$w_{n+1}\in F(w_n)$ for all $n$.
	%\item Let $\hat S'$ be the subset of $S'$ consisting of vectors based at non-vertex points on the edges.  $\hat S'C$, 
	%$\hat S'_xC$, $\hat S'_x$ are all defined similarly. 
	%\item Let $V \subset W$ be the sequences such that $v_n\in \hat S'$ for all $n$.
	\item Let $\sigma$ be the left shift on $W$.
	\item Let $t_v$ be the length of the geodesic segment in $C$ generated by $v$.
%	\item Let $\mathbb{P}(X)$ be the set of germs of geodesics in $X$.  This is an analogue to the projective tangent bundle which allows 
%	for branching. Over the interior of chambers it is the usual projective tangent bundle.  At branch points which are not vertices, these germs can be 
%	described using pairs of vectors in $S'$. Note that $\mathbb{P}(X)$ is compact.
\end{itemize}


%TO DO: PICTURES - DC


The geodesic flow on $G(\tilde X)$ is $g_t(\gamma(s)) = \gamma(s+t)$. It can also be realized by the suspension flow 
over $\sigma: W \to W$ with suspension function $g((w_n)) = t_{w_0}$.  Denote the suspension flow by $f_t: W_g \to W_g$ where 
\[W_g=\{((w_n),s): 0\leq s \leq g((w_n))\}/[((w_n),g((w_n)))\sim(\sigma((w_n)),0)]\]
and $f_t((w_n),s) = ((w_n),s+t)$. An explicit conjugacy between the suspension flow and the geodesic flow on the space $G'(X)$ of all geodesics which do not hit a vertex is as follows: $h: G'(X) \to W_g$ by $h(\gamma(t)) = ((w^\gamma_n), t^\gamma)$ where $(w^\gamma_n)$ is the trajectory of $\gamma$ through $S'$ indexed so that $w_0$ is $\dot\gamma(-t^\gamma)$ for $t^\gamma$ the smallest $t\geq 0$ for which $\dot\gamma(-t^\gamma)$ belongs to $S'$.


\begin{rem}
The spaces $G(\tilde X)$ and $G(X)$ are independent of the choice of metric on $X$. In contrast, 
the spaces $W$ and $G'(X)$ do a priori depend on the choice of metric.
\end{rem}














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\subsection{Liouville measure.}

We also want an analogue of Liouville measure.  We use the one constructed in \cite{ball_brin}.  On $S'$ define $\mu$ by 
\[ d\mu(v) = \cos\theta(v) d\lambda_x(v)dx \]
where $\theta(v)$ is the angle between $v$ and the normal to the edge it is based at, $\lambda_x$ is the Lebesgue measure 
on $S'_x$ and $dx$ is the volume on the edge.  This measure is invariant under $I$ by an argument well known from billiard 
dynamics (see e.g. \cite{CFS}). %The push-forward of $\mu$ via the inclusion map $\hat S' \to S'$ of $\mu$ induces a measure on $S'$.


Consider $W$ as the state space for a Markov chain with transition probabilities

\begin{equation*}
	p(v,w) = \left\{
	\begin{array}{cc}
		\frac{1}{|F(v)|} & \mbox{if } w\in F(v) \\
		0 & \mbox{else.}
	\end{array}\right.
\end{equation*}
Ballmann and Brin prove that $\mu$ is a stationary measure for this Markov chain (\cite{ball_brin} Prop 3.3) and hence $\mu$ 
induces a shift invariant measure $\mu^*$ on the shift space $W$.  %Again via the inclusion $V\subset W$ we can view $\mu^*$ as a shift-invariant measure on $W$ as well. 
Under the suspension flow on $W_g$, $\mu^*\times dt$ is invariant. Using the conjugacy $h$, pull back this measure to $G'(X)\subset G(X)$ and denote the resulting geodesic flow-invariant measure induced on $G(X)$ by $L$ (or $L_g$ if we want to specify the underlying metric $g$). As Ballmann and Brin remark, $\mu\times dt$ is the Liouville measure on the interior of each chamber $C$, so $L$ is a natural choice as a Liouville measure analogue on $G(X)$. %Note that we have slightly adjusted this from Ballmann and Brin's construction, by allowing the geodesics which meet vertices as a zero measure set.



We close this section with a quick remark about geodesics along walls.  In $\mathcal{M}^v_*$, every wall is geodesic and every geodesic tangent to an edge remains in $X^{(1)}$, by Corollary \ref{metric-angles-equal-comb-angles}. 
Without the non-singular vertices condition (i.e. in $\mathcal{M}_{neg}$) it is possible that a geodesic might spend some time along a wall, leave it, 
then return to it or another wall later.  This sort of behavior introduces some technical issues in the argument of section \ref{vol_conject}, which we will note there. For now, we make the following observation: 

\begin{prop}\label{prop:walls zero}
Let $g$ be a metric in $\mathcal{M}_{neg}$. Let $T$ be the set of geodesics which are tangent to a wall at some point.  Then $L_g(T)=0$.
\end{prop}


\begin{proof}
%This follows from the local expression for $d\mu$.  Indeed, it is easy to see that if the $\mu$ measure of those vectors tangent to walls is zero, then $L_g(T)=0$.  We had $d\mu(v)=\cos\theta(v)d\lambda_x(v)dx$ where $\lambda_x$ is the Lebesgue measure on the set of geodesics $\hat S'_x$.  Since only two of these geodesics are tangent to the wall, we get the desired result.

By a standing assumption, each edge in $X$ is geodesic. Thus, any geodesic which is tangent to a wall at some point will hit a vertex. These geodesics are omitted in the construction of $L_g$, and hence form a zero measure set when we think of $L_g$ as a measure on all of $G(X)$.
\end{proof}



%\todo{Check this argument? If you have an edge terminating at a vertex with large cone angle, can't it then extend past the vertex to 
%a large (positive measure) family of geodesics?} -- This is ok. such a family of geodesics would still have zero measure since it is codimension 1 in any chamber. DC

%\vskip 10pt

%\subsection{Flow boxes}


%We need flow boxes as a technical element in several of our proofs (see \cite{bonahon_ends} for a definition in the surface case). A \emph{flow box} $B$ centered on a geodesic segment $\lambda$ is a subset of $\mathbb{P}(X)$ which includes the segment $\lambda$ and is built as follows. Draw an $H$ shape on $X$ where $\lambda$ is the horizontal bar and the two vertical bars are geodesic segments transverse to $\lambda$ and the entire configuration is contained in a single apartment. Choose the vertical segments small enough that every geodesic segment which joins the two vertical bars and is homotopic to a path in the $H$ configuration meets the vertical bars transversely. $B$ consists of the $\mathbb{P}(X)$ images of these geodesic segments joining the vertical bars of the $H$. 
%For technical reasons, we also require that the boundary of $B$ not contain any segment of a closed geodesic. 
%Each flow box is foliated by geodesic segments; we call the foliation $\mathcal{F}$.

%\begin{exmp}
%If $\lambda$ joins two sides of a chamber, a flow box around $\lambda$ could be the set of all geodesic segments joining those two sides.  This sort of flow box is easily encoded in $W$.
%\end{exmp}


%The following proposition is a consequence of the compactness of $\mathbb{P}(X)$.


%\begin{prop}
%There exists a finite cover of $\mathbb{P}(X)$ by flow boxes.
%\end{prop}

%\begin{proof}
%This would follow directly from compactness if all flow boxes were open sets. However, flow boxes which cross an edge are not open. Let $e$ be an edge in $X$. Each pair of chambers $C_1, C_2$ meeting at $e$ form a subset of an apartment, and in such an apartment, the flow boxes are open sets. Then by compactness of $\mathbb{P}(C_1 \cup C_2)$, there is a finite cover of this set by flow boxes. Repeating this for the finitely many distinct chamber pairs meeting at $e$, and then for the finitely many edges in $X$ gives us a finite cover of the set of geodesic germs in $\mathbb{P}(X)$ based at non-vertex edge points. A similar argument extends this finite cover to those germs based at vertices. The complement of this finite cover is a closed subset consisting entirely of germs based at points in the interior of chambers. Over this closed set, flow boxes are open, so the result now follows by compactness.
%\end{proof}

\vskip 10pt







\subsection{Geodesic currents}

\begin{defn}
Let $\mathcal{G}(\tilde X_i)$ denote the space of (un-parametrized and un-oriented) geodesics in $(\tilde X, \tilde g_i)$.  A \emph{geodesic current} on $X = \tilde X/\Gamma$ is a positive Radon measure on $\mathcal{G}(\tilde X_i)$ which is $\Gamma$-invariant and cofinite (recall that a Radon measure is a Borel measure which is both inner regular and locally finite).
\end{defn}

\begin{exmp} The following are geodesic currents on a compact Fuchsian building quotient $X$ which will play a role in our later proofs:
\begin{itemize}
\item Any geodesic flow-invariant Radon measure on $G(\tilde X)/\Gamma$ induces a geodesic current on $X$, so $L_g$ induces
the Liouville current, also denoted $L_g$.  
\item For any primitive closed geodesic $\alpha$ in $X$, the Dirac mass $\mu_\alpha$ supported on the $\Gamma$-orbit of $\tilde \alpha$ 
is a geodesic current.  
\item For a non-primitive closed geodesic $\beta = \alpha ^n$, one can scale the current $\mu_\alpha$ associated to its primitive 
representative $\alpha$ by the multiplicity $n$, i.e. define $\mu_\beta := n\cdot \mu_\alpha$.  
\end{itemize}
\end{exmp}


Let $\mathcal{C}(X)$ denote the space of geodesic currents.  We equip it with the weak-$^*$ topology, under which it is complete (see, e.g. 
Prop. 2 of \cite{bonahon_geometry}).


\begin{prop}\label{density}
Let $\mathcal{C}\subset \mathcal{C}(X/\Gamma)$ be the set of currents which are supported on a single closed geodesic.  (I.e., it consists of all positive multiples of the $\mu_\alpha$ described above.)  Then $\mathcal{C}$ is dense in $\mathcal{C}(X/\Gamma)$.  
\end{prop}


\begin{proof} In \cite[Theorem 7]{bonahon_negative}, Bonahon establishes the analogous property for geodesic currents on 
$\delta$-hyperbolic groups, with a proof given in \cite[Section 3]{bonahon_negative}. 
Bonahon's argument makes use of the Cayley graph $Cay(G)$ of $G$, but only relies on negative curvature properties of the Cayley
graph -- the group structure plays no role in the proof. A careful reading of the arguments shows that it applies verbatim in our setting.
\end{proof}


Since $X_0$ and $X_1$ are compact locally $CAT(-\epsilon^2)$ spaces with the same fundamental group, there is a 
$\Gamma$-equivariant homeomorphism $\partial^\infty \tilde X_0\to \partial^\infty \tilde X_1$ induced by the identification of $\partial^\infty \tilde X_i$ with $\partial^\infty\Gamma$. Since the space of 
geodesics is given by $\mathcal{G}(\tilde X_i) = [(\partial^\infty \tilde X_i \times \partial^\infty \tilde X_i)\setminus \Delta]/[(\gamma_1,\gamma_2)\sim (\gamma_2,\gamma_1)]$, 
there is an induced homeomorphic orbit equivalence $\varphi:\mathcal{G}(\tilde X_0)/\Gamma \to \mathcal{G}(\tilde X_1)/\Gamma$. Thus, $\Gamma$-invariant Radon measures on $\mathcal{G}(\tilde X_1)$ of cofinite mass can be pulled back via $\varphi$ to obtain $\Gamma$-invariant Radon measures on $\mathcal{G}(\tilde X_0)$ of cofinite mass. Similarly, currents can be pushed forward via $\varphi$. We fix the following notation:

\begin{defn}\label{pull-push}
For a current $\alpha \in \mathcal{C}(X_0)$, the push-forward of $\alpha$ under $\varphi$ is $\varphi_*\alpha$, defined by $\varphi_*\alpha(A)=\alpha(\varphi^{-1}(A))$. For a current $\beta \in \mathcal{C}(X_1)$, the pull-back of $\beta$ under $\varphi$ is $\varphi^*\beta$, defined by $\varphi^*\beta(B)=\beta(\varphi(B))$.
\end{defn}

The following is a straightforward consequence of the $\Gamma$-equivariance of $\varphi$.

\begin{lem}\label{lemma:pull-closed}
Let $[\gamma]$ be any element of $\Gamma = \pi_1(X)$, and let $\gamma_i \in \mathcal{C}(X_i)$ be the current supported on the closed $g_i$-geodesic representing $\gamma$. Then $\varphi_*\gamma_0 = \gamma_1$ and $\varphi^*\gamma_1=\gamma_0$.
\end{lem}


If we equip $\mathcal{C}(X_i)$ with the weak-* topology, then the following is also an easy consequence of the definitions:


\begin{lem}
If $\alpha_n \to \alpha$ in $\mathcal{C}(X_0)$, then $\varphi_*\alpha_n \to \varphi_*\alpha$ in $\mathcal{C}(X_1)$. If $\beta_n\to \beta$ in $\mathcal{C}(X_1)$, then $\varphi^*\beta_n\to \varphi^*\beta$ in $\mathcal{C}(X_0)$.
\end{lem}



The Corollary below will be used for a technical step in the proof of Theorem \ref{thm:MLS-determines-volume}:



\begin{lem}
Let $\partial^\infty f: \partial^\infty(\tilde X,\tilde g_0) \to \partial^\infty(\tilde X, \tilde g_1)$ denote the $\Gamma$-equivariant homeomorphism induced by identifying $\partial^\infty \tilde X_i$ with $\partial^\infty \Gamma$, and $\partial^\infty \Phi:\partial^\infty(\tilde X,\tilde g_0) \to \partial^\infty(X, \tilde g_1)$ the map induced by the lift of the identity map $X\to X$. Then $\partial^\infty f = \partial^\infty \Phi$. 
\end{lem}

\begin{proof}
Fix a fundamental domain $\mathcal{F}$ for the $\Gamma$-action on $(\tilde X,\tilde g_0)$. Any geodesic ray $\eta$ in $(\tilde X,\tilde g_0)$ can be encoded by a sequence $(\gamma_i\cdot \mathcal{F}_i)$ through which it passes; the sequence $(\gamma_i)$ approaches the point in $\partial^\infty \Gamma$ corresponding to $\eta(+\infty)$. The sequence $(\gamma_i\cdot \Phi(\mathcal{F}))$ approaches $\partial^\infty\Phi(\eta(+\infty))$. Likewise, $\partial^\infty f(\eta(+\infty))$ is the point in $\partial^\infty(\tilde X,\tilde g_1)$ corresponding to $(\gamma_i)$, namely the point which $(\gamma_i\cdot\Phi(\mathcal{F}))$ approaches. Thus $\partial^\infty f(\eta(+\infty))=\partial^\infty\Phi(\eta(+\infty))$.
\end{proof}


\begin{cor}\label{wall-corollary}
The map $\varphi:\mathcal{G}(\tilde X_0)/\Gamma \to \mathcal{G}(\tilde X_1)/\Gamma$ sends walls to walls.
\end{cor}

\begin{proof}
As noted in the proof of Theorem \ref{thm:MLS-rigidity-special-case}, $\partial^\infty \Phi$ maps endpoints of walls to endpoints of walls (see \cite[Lemma 3.11]{X}). By the previous Lemma, so does $\partial^\infty f$. Since $\partial^\infty f$ defines $\varphi$, $\varphi$ sends the geodesic connecting the endpoints at infinity of some wall in $(\tilde X,\tilde g_0)$ to the endpoints at infinity of the same wall in $(\tilde X,\tilde g_1)$. Our vertex regularity condition, together with Corollary \ref{metric-angles-equal-comb-angles} implies that these geodesics are the walls themselves.
\end{proof}







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\section{Intersection pairing on Fuchsian buildings}\label{sec:pairing}



The key tool in the proof of Theorem \ref{thm:MLS-determines-volume}, as in Otal's original work on MLS rigidity and Croke and Dairbekov's work on MLS and volume is the intersection pairing for geodesic currents. This is a finite, bilinear pairing 
on the space of currents, which recovers the intersection number for geodesics when the currents in question are Dirac measures on closed geodesics, and can also recover lengths of closed geodesics and the total volume of the space. For surfaces it is defined by

\[ i(\mu,\lambda) = (\mu\times \lambda)(DG(X)) \]
where $DG(X)$ is the set of all transversally intersecting pairs of geodesics on $X$.


The main problem in extending this tool to the building case is the fact that $DG(X)$ is not naturally defined for 
buildings. For a surface, transverse intersection of geodesics is detected by linking of their endpoints in the circle 
$\partial^\infty\tilde S$. This is no longer the case for buildings, and one can imagine a pair of geodesics which 
intersect in $(\tilde X,g_0)$ but whose images under $\varphi$ do not intersect in $(\tilde X,g_1)$ due to the 
branching of the building.


Therefore, we must introduce an `adjusted' version of both $DG(X)$ and the intersection pairing and prove 
that it retains enough of the necessary properties of $i(-,-)$ for our purposes. We begin by examining to what 
extent we can detect intersections of geodesics in a building. Before doing so we make the following definition:



\begin{defn}
We say that two geodesics $\gamma_1$ and $\gamma_2$ \emph{intersect transversally} if there exists an 
apartment $A$ such that the restrictions of $\gamma_1$ and $\gamma_2$ to $A$ intersect transversally in 
their interiors. Note in particular that each $\gamma_i\cap A$ is a non-degenerate segment. We 
say that they \emph{intersect in a point} if $\gamma_1\cap \gamma_2$ is a single point.
\end{defn}

Note that two geodesics intersecting in a point in the interior of a chamber automatically intersect transversally. 
If $\gamma_1$ and $\gamma_2$ intersect transversally, then there is a well-defined angle between them, up to the 
symmetry about $\frac{\pi}{2}$.


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\subsection{A lemma on links}

Let $I_p$ denote the interval graph with $p$ segments. Let $S^1_q$ denote the circle graph with $q$ segments. 


\begin{lem}\label{linklemma}
Let $G$ be a thick, generalized $m$-gon. Then any embedding of $I_p$ into $G$ for $p\leq 2m$ can be extended to an embedding of $S^1_q$ into $G$ for some $q\geq p$.
\end{lem}

\begin{proof}
If $p\leq m+1$, then the embedding of $I_p$ lies in an apartment in $G$; the apartment itself is an extension of $I_p$ to an embedding of $S^1_{2m}$.

Assume then that $p>m+1$. Let $v_0$ and $v_p$ be the endpoints of the embedding of $I_p$ in $G$. Let $v_m$ be the vertex at distance $m$ from $v_0$ along this embedding. Let $a$ denote the segment of the embedding from $v_0$ to $v_m$. Let $b$ denote the segment of the embedding from $v_m$ to $v_p$; it has length $p-m$ with $m\geq p-m>1$.  

The length $m$ segment in the embedding from $v_0$ to $v_m$ can be extended to an apartment $A_1$ by connecting $v_m$ and $v_0$ by a length $m$ segment $c$. Since the $m$-gon is thick, we may assume that the segment of $c$ leaving $v_n$ does not agree with the segment of $b$ leaving $v_m$. Let $c'$ be the initial segment of $c$ of length $m+1-(p-m)$ so that the concatenation of $b$ with $c'$ is a path of length $m+1$. Note that $b\cap c = \{v_m\}$, else a loop of length $<2m$ exists, which is not allowed.  As any path of length $m+1$ can be extended to an apartment, there must exist a segment $d$ of length $m-1$ connecting $v_p$ to the end of $c'$ and forming apartment $A_2$. (See Figure \ref{fig:polygon}.)


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\begin{figure}[h]
\caption{Segments in the proof of Lemma \ref{linklemma}.}\label{fig:polygon}
\end{figure}

\end{center}

Note that if $d\subset (c\setminus c')\cup a$ but $d$ is not contained in $c\setminus c'$ (which is ruled out as $b\cap c=\emptyset$), then apartments $A_1$ and $A_2$ agree along a path of length $m+1$, which would imply they are the same apartment, and contradict our choice of the segment $c$ as branching away from $b$ at $v_m$. 

Next we note that $d\cap a =\emptyset$, for if not, let $d'\subset d$ be the segment of $d$ between its final intersection 
with $a$ and its subsequent intersection with $c$. The segments $a$ and $c$ form an embedded circle of length $2m$; 
$d'$ is a chord of length $<m$ connecting two points on this circle. But then there must be a loop of length $<2m$ 
present in this configuration, which is a contradiction. Finally, since $d\cap a =\emptyset$, the concatenation of $a, b, d$ 
and $c\setminus c'$ (canceling any portion of $d$ 
that lies along $c\setminus c'$) provides the desired embedded circle extending $I_p$.

\end{proof}


%\begin{rem}
%If we assume that $p<2m$, we may slightly adapt the argument above by letting $b\cup c'$ have length $m$. Then there are at least two ways to choose a length-$m$ segment $d$ connecting $v_p$ to the end of $c'$ which are disjoint apart from their endpoints. At least one of these does not go through $v_0$ and provides us with an embedded $S^1$ of length strictly greater than $2m$. We will use this adjustment for one technical point in the proof of Lemma \ref{lem:strip fill} below.
%\end{rem}





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\subsection{Intersections with walls and edges}


Since $\varphi$ sends walls to walls via the combinatorial isomorphism (see Corollary \ref{wall-corollary}),
if $w_1\cap w_2 \neq \emptyset$, then $\varphi(w_1)\cap\varphi(w_2)\neq \emptyset$. A more 
interesting result is the following:


\begin{prop}\label{geodwall}
Let $\gamma$ be any geodesic and $w$ a wall. Suppose that $\gamma$ and $w$ intersect transversally. Then 
$\varphi(\gamma)$ and $\varphi(w)$ intersect.
\end{prop}

Observe that, a geodesic intersecting a wall in the interior of an edge automatically intersects the wall transversely. 
Thus the proposition immediately implies

\begin{cor}
If the geodesic $\gamma$ and wall $w$ intersect in a {\em non-vertex} point, then $\varphi(\gamma)$ and $\varphi(w)$
must also intersect.
\end{cor}

%\begin{proof}
%Consider the apartment $A$ containing $\gamma$. Since $w$ intersects $\gamma$ in the interior of an edge $e$, the
%edge $e$ must lie inside $A$. Hence $A$ is an apartment witnessing the transverse intersection of 
%Choose the two chambers adjacent to $w$ surrounding the intersection can be chosen to show that $\gamma$ and $w$ intersect transversally.
%\end{proof}


%\begin{lem}\label{wallfill}
%Let $w$ be a wall and $\gamma_+$ a geodesic ray such that $\gamma_+\cap w$ is the initial point of $\gamma_+$. Suppose that there is an apartment $A$ such that the restrictions of $\gamma_+$ and $w$ to $A$ intersect transversally in the interior of $w|_A$. Then there exists a totally geodesic half-space bordered by $w$ containing $w$ and $\gamma_+$.
%\end{lem}


%\begin{rem}
%Note that the half-space constructed in this lemma is not necessarily a half-apartment. It may contain singular points -- vertices $v$ around which the total angle of the half-space is $>2\pi$.
%\end{rem}


%The proof of Lemma \ref{wallfill} consists of an inductive argument, building the desired half-space using `strips.'


\begin{defn}\label{defn:tot geod strip}
A \emph{totally geodesic strip} is any union of chambers of $\tilde X$ which is convex, on which the induced metric is locally $\mathrm{CAT}(-1)$, and which is homeomorphic to $[0,1]\times \mathbb{R}$.  A \emph{totally geodesic plane}
is similarly defined, but is homeomorphic to $\mathbb R^2$.
\end{defn}


The $\mathrm{CAT}(-1)$ condition means that, around any vertex in the interior of a totally geodesic strip $S$, the total angle 
is $\geq 2\pi$. If a vertex $v$ lies on $\partial S$, i.e. the image of $\{0,1\}\times \mathbb{R}$ under the homeomorphism, 
then the image of $S$ in $lk(v)$ consists of an embedded sequence of some number of consecutive edges (i.e.
an embedded combinatorial path in $lk(v)$). Let $l(v)$ denote this number of edges. If $lk(v)$ is a generalized $m_v$-gon, then convexity at $v$ is equivalent to $l(v) \leq m_v$. Convexity of the strip then reduces to this condition at each boundary vertex.


The proof of Proposition \ref{geodwall} relies on the following technical lemma.  In it, we show that a convex totally geodesic strip can be thickened to another convex totally geodesic strip, subject to a certain constraint on how that thickening occurs near a particular edge of $\partial S$.







\begin{lem}\label{lem:strip fill}
Let $S$ be a convex totally geodesic strip. Fix an edge $e$ in one component $B$ of $\partial S$ with endpoints 
$v_0$ and $v_1$. Let $\hat S$ be the union of $S$ with a sequence of 
$t\leq m_{v_1}$ chambers $C_1, \ldots, C_t$, with the property that (i) $C_1\cap B = e$, (ii) for 
$i\geq 2$, $C_i \cap B =\{v_1\}$, and (iii) each $C_i\cap C_{i+1}$ is an edge. (See Figure \ref{fig:schematic}.)
Then there exists a convex totally geodesic strip $S'$ containing $\hat S$ such that 
$B\cap \partial S' = \emptyset$.
\end{lem}


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\begin{figure}[h]
\caption{$\hat S$, extended to $S'$ with the new boundary of $S'$ in dashed lines, as in Lemma \ref{lem:strip fill}.}\label{fig:schematic}
\end{figure}

\end{center}





\begin{proof}

We will prove this Lemma under two cases, first the case where the chambers have at least four sides, and then the case that they are triangles. The latter case requires some slight additions to the argument.


\vspace{.5cm} 
\noindent\textbf{Case 1: Chambers have at least four sides.}


Enumerate the vertices in $B$ in order as $\{v_i\}_{i\in\mathbb{Z}}$, agreeing with the specification of $v_0$ and $v_1$. Convexity of the strip guarantees that this is possible.



Consider the image of $\hat S$ in $lk(v_1)$. It is an interval of $\leq 2m_{v_1}$ edges, since at most $m_{v_1}$ edges 
are contributed by $S$ and an additional $t\leq m_{v_1}$ edges come from the chambers added to $S$ to 
form $\hat S$. By Lemma \ref{linklemma}, this interval can be extended to an embedded $S^1$ in $lk(v)$. Glue 
chambers to $\hat S$ adjacent to $v_1$ according to this $S^1$; call the result $\hat S_1$.

Now consider $v_2$. Of the chambers added to $\hat S$ in forming $\hat S_1$, exactly one is adjacent to $v_2$, specifically the added chamber corresponding to the edge in $S^1\subset lk(v_1)$ which completes the $S^1$. None of the other added chambers can be adjacent to $v_2$ because chambers are convex, and a second chamber attached to $v_1$ and $v_2$ not along the edge between them would provide a second geodesic path between the two vertices, a contradiction. Thus, the image of $\hat S_1$ in $lk(v_2)$ is an interval of $\leq m_{v_2}+1<2m_{v_2}$ edges. As in the previous step, apply Lemma \ref{linklemma} and glue new chambers accordingly to obtain $\hat S_2$. Via the same argument as above, the image of $\hat S_2$ in $lk(v_3)$ is an interval of $\leq m_{v_3}+1<2m_{v_3}$ edges.

Continue in this fashion, gluing new chambers to vertices $v_i$ with $i\to \infty$. Then run the same process on $v_i$ with $i\to -\infty$. Call the resulting union of chambers $S'$. By construction, $B\cap \partial S' = \emptyset$. It also is clear from the construction that $S'$ is locally $\mathrm{CAT}(-1)$, since its image in the link of any interior vertex $v$ is a circle with $\geq 2m_v$ edges. That it is homeomorphic to $[0,1]\times \mathbb{R}$ will follow once we check the convexity condition on the vertices of $\partial S'$, since convexity implies the vertices along the boundary components can be ordered linearly as noted above, and hence that there are only two boundary components.


Suppose that $C_1$ and $C_2$ are chambers attached to $v_i$ and $v_j$ with $|i-j|\geq 2$. We claim that $C_1\cap C_2 = \emptyset$. Indeed, from $|i-j|\geq 2$ it follows that there exists a chamber $C_3$ incident to an edge $e$
along $\partial S$, and lying between $C_1$ and $C_2$. Consider the two edges $e_1, e_2$ of $C_3$ which are 
adjacent to $e$, and form the sets $W_i$ consisting of all walls passing through $e_i$. Since the chamber $C_3$
has at least four sides, we have $e_1\cap e_2=\emptyset$. The chamber $C_3$ is metrically a convex hyperbolic
polygon, so it contains a geodesic segment $\eta$ which is orthogonal to both $e_1$ and $e_2$. 
This implies $W_1\cap W_2=\emptyset$. Finally, we note that each $W_i$ separates $\tilde X$ into two 
connected components. By construction, we have that $C_1$ and $C_2$ lie in different connected components
of (each) $\tilde X \setminus W_i$, and so indeed, $C_1\cap C_2=\emptyset$. 


%Suppose not, and suppose that $v \in C_1\cap C_2$ is the vertex in the intersection which is at shortest combinatorial distance from both $v_i$ and $v_j$. There is a triangle $T$ of edge paths connecting $v_i, v_j$ and $v$ and by convexity of $S$ and $C_i$, the geodesics connecting these three vertices lie in $S\cup C_1 \cup C_2$. This triangle is at least four edges long. 

%Let $p$ be the once-broken geodesic path from $v_i$ to $v_j$ through $v$ and $\gamma$ the geodesic path from $v_i$ to $v_j$. There is a homotopy of paths from $p$ to $\gamma$, so there must be some finite union of chambers $T^*$ which is simply connected and has boundary $T$. But this is impossible, by the following argument. Let $C_3$ be any chamber in $T^*$ adjacent to an edge $e$ between $v_i$ and $v_k$. Take two non-adjacent sides of $C_3$, neither of which are $e$. Extend these to geodesic edge paths in $T^*$. They do not intersect, and $C_1$ lies to the outside of one, $C_2$ to the outside of the other. Hence $C_1\cap C_2 = \emptyset$.


%First, if possible, pick some vertex $v_k$ on the edge path between $v_i$ and $v_j$ where the edge path is not geodesic. The angle on the $T^*$-side at $v_k$ is $>\pi$. Thus we can extend the two edges in $T$ adjacent to $v_k$ to geodesic paths through $T^*\cup C_1\cup C_2$ which lie in the 1-skeleton of $\tilde X$. But $C_1$ must lie to the outside of one such path, and $C_2$ to the outside of the other such edge path, so $C_1$ and $C_2$ in fact do not intersect. A similar argument works if the edge path $p$ has more than two edges.


%The only case that remains to check is when the side of $T$ between $v_i$ and $v_j$ is geodesic and $p$ is a union of two edges. Let $C_3$ be any chamber in $T^*$ adjacent to the edge $e$ between $v_i$ and $v_k$. Take two non-adjacent sides of $C$, neither of which are $e$. Extend these to geodesic edge paths in $T^*$. They do not intersect, and $C_1$ lies to the outside of one, $C_2$ to the outside of the other. Hence $C_1\cap C_2 = \emptyset$.


Now that we know $C_1$ and $C_2$ do not intersect, we see that a vertex $v\in \partial S'$ can belong to at most two chambers attached to $v_i$, at most two attached to $v_{i+1}$ and no chambers attached to other vertices in $\partial S$. The only case in which the bound of two is achieved for both vertices is when the chamber is a triangle, which will be dealt with in Case 2. In fact, for chambers with at least four edges, if two chambers adjacent to $v_i$ meet at $v$, then no chamber adjacent to $v_{i+1}$, besides, perhaps, one of these two chambers, contains $v$. Thus, for all vertices in $\partial S'$, $l(v)\leq 2$, and convexity holds. This finishes the proof of Case 1.




\vspace{.5cm}
\noindent\textbf{Case 2: The chambers are triangles.}


%If $n_{v_k}>2$, then the argument above works again. If not, $n_{v_k}=2$ and there is a geodesic path in the 1 skeleton of $T^*\cup C_1\cup C_2$ starting from $v_k$ perpendicular to $\gamma$. If $|i-j|>2$ we can find two such paths, both perpendicular to the geodesic path from $v_i$ to $v_j$ which again must separate $C_1$ and $C_2$.  If $|i-j|=2$, there must be a single edge connecting $v_k$ to $C_1\cap C_2$ and separating $T^*$ into the union of two triangular regions.

We repeat the argument above, but need to adjust slightly to ensure convexity of $S'$. The arguments in Case 1 apply except for the argument proving $C_1\cap C_2 =\emptyset$. A very similar argument works except for the case where $|i-j|=2$. Then there is a triangle $T$ bounded by two edges along $w$ and one edge each from $C_1$ and $C_2$. The pair of edges between $v_i$ and $v_j$ along $w$ must form a geodesic segment, otherwise an argument similar to Case 1 works again, extending each of these edges into a collection of walls $W_i$. Furthermore, if the combinatorial angle at the vertex $v_k$ between these edges is not $\pi/2$, then there are two distinct edges from $v_k$ which can be extended to collections $W_i$, allowing Case 1's argument again. Note that by Feit-Higman (\cite{FH}) the combinatorial angles of the chamber are $(\pi/2, \pi/3, \pi/8)$ or $(\pi/2, \pi/k, \pi/l)$ with $k,l\in\{4,6,8\}$ but not both $=4$.


Combining this with the arguments of the last paragraph of Case 1, we see that in the triangle case, we may have vertices in $\partial S'$ with $l(v) = 3$ or 4, the latter only in the case that $|i-j|=2$ and the combinatorial angle at $v_k$ is $\pi/2$. The only problems for convexity at such a vertex arise when the combinatorial angle there if $\pi/2$ or $\pi/3$. Figure \ref{fig:convex} shows the only two non-convex possibilities. 


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\begin{figure}[h]
\caption{Filling in non-convex regions in Lemma \ref{lem:strip fill}.}\label{fig:convex}
\end{figure}

\end{center}

In each of these possible configurations we see in the dashed additions to the figure, that a few extra chambers can be added (using Lemma \ref{linklemma}, or the fact that any $m_v+1$ adjacent chambers around a vertex can be completed into a circuit of length $2m_v$) to remove the non-convexity at the problematic vertex. This proves Case 2.



\end{proof}




\begin{proof}[Proof of Proposition \ref{geodwall}]
%The statement is trivial if $\gamma_+\subset w$. If $\gamma_+$ lies along $w$ for some time, then branches away, we can replace it with just the branching segment and reduce to the case that $\gamma_+\cap w$ is a single point.

We may assume that $\gamma$ is not a wall, else the statement follows from the fact that $\varphi$ agrees with the combinatorial isomorphism along walls.


Let $\mathcal{C}$ be the union of a minimal set of chambers which covers $\gamma$, which includes the chamber(s) witnessing the transversal intersection of $\gamma$ and $w$, and such that any chamber in the collection has at least two edges which belong to another chamber in the collection. 


Let $\mathbf{C}$ be the 1-skeleton of $\mathcal{C}$. The boundary of $\mathcal{C}$ consists of two embedded copies of $\mathbb{R}$ with marked points at the vertices. At each marked vertex the number $l(v)$ indicates the combinatorial length of the image of $\mathcal{C}$ in $lk(v)$. We have $l(v)\leq m_v+1$  for all $v$ since the geodesic $\gamma$ lies in some apartment containing $v$ and can only intersect each wall through $v$ once (see Figure \ref{fig:gamma}). 

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\put(20,130){$\gamma_+$}


\end{picture}

\begin{figure}[h]
\caption{A geodesic passing near a vertex.}\label{fig:gamma}
\end{figure}

\end{center}





%It remains only to check the vertex $v^*$ along $w$ where $\mathbf{B}$ departs from $w$. Here, as Figure \ref{fig:wall} shows, $d(v^*)\leq m+2$. Note that $w$ and $\gamma_+$ may not lie in the same apartment in this figure -- $\gamma_+$ may branch away at some point after crossing the sector labeled $S$.


%\begin{center}

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%\begin{picture}(300,300)(-150,-150)
	
	
%\put(0,0){\circle*{5}}
%\put(22,-18){$v^*$}
%
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%\qbezier(100,100)(0,0)(-100,-100)

%\qbezier(-80,00)(0,5)(120,70)

%\put(120,50){$\gamma_+$}
%\put(120,-13){$w$}
%\put(-120,30){$S$}


%\end{picture}

%\begin{figure}[h]
%\caption{A geodesic ray $\gamma_+$ intersecting a wall near the vertex $v^*$.}\label{fig:wall}
%\end{figure}

%\end{center}



%In the first situation described above, the image of $\mathbf{B}$ in $lk(v)$ is a pair of antipodal points. In the second, the image is an embedded interval of length $\leq m+1$. In the third, $lk(v)$ is the union of a single point together with an embedded interval of length $\leq m$.


We now want to use Lemma  \ref{lem:strip fill} to extend $\mathcal{C}$ to a totally geodesic containing $\gamma$ and $w$.


First, consider $w$ as a convex totally geodesic strip of zero width. Let $v_0$ and $v_1$ be the vertices in $\mathcal{C} \cap w$ nearest $\gamma\cap w$. There are at most $l(v_1)$ chambers in $\mathcal{C}$ adjacent to $v_1$, but since $\gamma$ crosses $w$, we may restrict our attention to chambers containing a ray $\gamma_+$, of which at most $l(v_1)-1\leq m_{v_1}$ are adjacent to $v_1$ and one is adjacent to $v_0$ (up to interchanging the two vertices). We can then apply Lemma \ref{lem:strip fill} to extend $w$ on this side to a wider convex totally geodesic strip.


(This argument needs to be slightly adjusted if $\gamma$ and $w$ intersect at a vertex, so that the chambers witnessing their transversal intersection -- which are in $\mathcal{C}$ -- form a full circuit around the vertex. In this case, declare this vertex to be $v_1$ and consider those chambers to be the first step in building the strip.)



In doing so, we make sure to fill using chambers in $\mathcal{C}$ whenever possible, so that the new strip covers as much of $\gamma_+$ as possible and $\gamma_+$ exits the new strip through its boundary. This will always be possible, because those chambers in $\mathcal{C}$ adjacent to the strip built so far, together with the strip itself, always have image in the link of any boundary vertex with length $\leq 2m_v$, allowing the application of Lemma \ref{linklemma}.



Continue to apply Lemma \ref{lem:strip fill} inductively. At each stage $\hat S$ consists of the strip produced so far, as well as any chambers in $\mathcal{C}$ adjacent to that strip on the $\gamma_+$ side. There are always at most $l(v)-1\leq m_v$ chambers from $\mathcal{C}$ added to $S$, since at least one chamber adjacent to $v$ has already been included in $S$. The induction produces a totally geodesic half-space containing $\gamma_+$ and $w$. Note that it need not be a half apartment -- there may be singular vertices with total angle $>2\pi$.


Run the same argument on the other side of $w$ to cover the other ray $\gamma_-$ and $w$. The fact that $\gamma$ and $w$ intersect transversally guarantees that the two half-spaces so produced together form a totally geodesic 
plane, call it $S^*$. Suppose it is not embedded. Then there are two distinct geodesics from $\gamma\cap w$ to a single point $x$ -- a point which is encountered twice in the process of constructing $S^*$. But this contradicts the fact that $\tilde X$ is $\mathrm{CAT}(-1)$.



Take any geodesic segment in $S^*$ emanating from $\gamma\cap w$. It is clear from the construction of $S^*$ that such a segment can be continued indefinitely, since it certainly traverses infinitely many chambers, and any $\max\{m_v\}$ chambers traversed consecutively covers a distance of at least $\rho$ along the segment, where $\rho$ is the minimum inradius of any chamber. Thus, the boundary at infinity of $S^*$ is an embedded circle in $\partial^\infty \tilde X$ containing $w(\pm\infty)$ and $\gamma(\pm\infty)$. The endpoints of $\gamma$ and $w$ are linked in this circle. This remains true after applying $\varphi$, which implies that $\varphi(\gamma)$ and $\varphi(w)$ must still intersect, as desired. 


%filling of the region between $w$ and $\gamma_+$ bounded by $\mathbf{B}$ with chambers so that we obtain a sector of a totally geodesic embedded half-space. We do this as follows. First, at the vertex $v^*$ where $w$ and the chambers covering $\gamma_+$ meet, connect the isolated point in $\mathbf{B}\subset lk(v)$ to its antipodal point in $\mathbf{B}$ with a segment of length $m$ disjoint from $\mathbf{B}\subset lk(v^*)$.  (Note that we are abusing notation a bit and denoting with the same symbol, $\mathbf{B}$, a subset of the 1-skeleton of $X$ and its image in a link.) We now have an embedded interval in $lk(v^*)$ of length $\leq 2m$. Using Lemma \ref{linklemma}, complete this to an embedded $S^1$ in $lk(v^*)$. Glue new chambers to $\mathcal{C}$ according to the edges in $lk(v^*)$ added in the application of Lemma \ref{linklemma}. Redefine $\mathcal{C}$ as the union of these new chambers with those previously present.


%After this gluing, the boundary component of $\mathcal{C}$, call it again $\mathbf{B}$, has changed as follows. The vertex $v^*$ we worked at above is no longer in $\mathbf{B}$. The vertices now in $\mathbf{B}$ which were previously adjacent to $v^*$ have had $1$ added to their degrees in the graph $\mathbf{B}$. Note that the new degrees are bounded above by $\leq m+2\leq 2m$. Vertices now in $\mathbf{B}$ which were not there previously have degree 2. The images of $\mathbf{B}$ in the links of these vertices have the same types as described above.


%This allows an inductive scheme for filling between $w$ and $\gamma_+$, since after each step as above $\mathbf{B}$ is still an embedded line with the same set of possible link types at its vertices. Specifically, first follow this process to attach chambers along the ray of $w$ in $\mathbf{B}$. Then, starting at an arbitrary point in the resulting $\mathbf{B}$, completely fill in both directions along $\mathbf{B}$ to fill a `strip' along $\mathbf{B}$. Then fill strip after strip in this way, completing the filling.



%At each step in this process, Lemma \ref{linklemma}'s use ensures that the image of $\mathcal{C}$ in each link is an embedded $S^1$. This link property shows that the embedded complex $\mathcal{C}^*$ which results from this process is locally totally geodesic. Take any geodesic segment in $\mathcal{C}^*$ emanating from $v$ with direction between $\gamma_+$ and $w$. It is clear from the construction that such a segment can be continued indefinitely, since it certainly traverses infinitely many chambers, and any $n$ chambers traversed consecutively covers a distance of at least $\rho$ along the segment where $\rho$ is the minimum inradius of any chamber. Thus, the boundary at infinity of $\mathcal{C}^*$ is an embedded interval in $\partial^\infty \tilde X$ connecting $w(\infty)$ and $\gamma_+(\infty)$, and so $\mathcal{C}^*$ is a sector of a totally geodesic embedding of $\mathbb{R}^2$.


%Do the same procedure on the other side of $\gamma_+$. Note that the two sectors created via this filling procedure intersect only along $\gamma_+$ for the following reason. If $x$ belonged to both sectors, there would be two distinct geodesic segments from $\gamma_+\cap w$ to $x$, one in each sector. This contradicts our negative curvature assumption. We thus obtain the totally geodesic half-space in the statement of the lemma.

%\end{proof}



%We now prove Proposition \ref{geodwall}.


%\begin{proof}[Proof of Proposition \ref{geodwall}]

%Write $\gamma$ as $\gamma_+ \cup \gamma_-$. Fill as in Lemma \ref{wallfill}, obtaining two totally geodesic half-spaces with common border $w$; they are disjoint except for $w$ by the argument noted at the end of the proof of Lemma \ref{wallfill}. The union of these is a totally geodesic embedded $\mathbb{R}^2$ in $\tilde X$ with boundary at infinity an embedded $S^1$. The endpoints of $\gamma$ and $w$ are linked in this circle. This remains true after applying $\varphi$, which implies that $\varphi(\gamma)$ and $\varphi(w)$ intersect as desired. 

\end{proof}



We need the following version of the above as well.


\begin{prop}\label{prop:vertex case}
Suppose that $\gamma$ and $w$ intersect transversally in a vertex $v$ at the end of edge $e$, with $e$ in the apartment $A$ in which $\gamma$ and $w$ intersect transversally. Then there is a second wall $w'$ through $e$ which branches away from $e$ at $v$, such that $\varphi(\gamma)$ and $\varphi(w')$ must intersect.
\end{prop}

\begin{proof}
We first show that there is an embedded $S^1$ in $lk(v)$ which contains the images of $\gamma$ and $w'$ in a linked fashion. 

In the link of $v$, let $e$ and $\bar e$ be the image of $w$, and $p, \bar p$ the image of $\gamma$. Pick any pair of opposite vertices $b,\bar b$ in the apartment containing $e, \bar e, p, \bar p$. Let $A'$ be an apartment which branches from $A$ along the wall from $b$ to $\bar b$, and let $e'$ be the vertex opposite $e$ in $A'$. The wall $w'$ will be chosen to go through $e$ and $e'$. (See Figure \ref{fig:S1}.)



To construct the desired embedded $S^1$, trace from $b$ through $A\cap A'$, passing through $p$ and $e$. We may assume, up to switching the role of $b$ and $\bar b$ that this curve hits $p$ first, then $e$. At $\bar b$ continue in $A$ through $\bar p$ and $\bar e$. In the remaining interval $[\bar e, b)$ there exists a vertex $q$ which is opposite to $q' \in A'$ such that $e'$ is between $q'$ and $b$ on $A'$. Follow $A$ to $q$, then, using thickness, follow a length $n$ a path from $q$ to $q'$ disjoint from all previous paths. Finally, complete the $S^1$ by following the short interval in $A'$ from $q'$ through $e'$ to $b$.

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\end{picture}

\begin{figure}[h]
\caption{Constructing the embedded $S^1$.}\label{fig:S1}
\end{figure}

\end{center}


Now from this embedded $S^1$ in $lk(v)$, we build a portion of $\mathcal{C}$ around $v$ by taking
the union of all chambers corresponding to the edges in the $S^1$, along with a totally geodesic strip along $w'$. 
Then, just as in Proposition \ref{geodwall}, we use Lemma \ref{lem:strip fill} to 
inductively build a totally geodesic plane $S^*$ inside $\tilde X$ which contains both $\gamma$ and $w'$. 
From the linking of the endpoints of 
$\gamma, w'$ in the boundary at infinity $\partial ^\infty S^* \cong S^1$ of the plane, it again follows
that the corresponding $\varphi(\gamma)$ and $\varphi(w')$ must intersect. This completes the proof.

\end{proof}



This proposition has the following consequence.


\begin{prop}\label{prop:edgeint}
Let $\gamma$ be a geodesic in $(\tilde X,g_0)$ and $e$ an edge of a wall $w$ such that $\gamma$ and $w$ intersect transversally and $\gamma\cap e \neq \emptyset$. Let $\varphi(e)$ be the edge in $(\tilde X,g_1)$ identified by the combinatorial isomorphism. Then $\varphi(\gamma)\cap \varphi(e) \neq \emptyset$.
\end{prop}



\begin{proof}
Find, using Propositions \ref{geodwall} or \ref{prop:vertex case}, a set of walls $\{w_i\}$ intersecting $\gamma$ with $\cap_i w_i=e$. Then, using those propositions, $\varphi(\gamma)$ and $\varphi(w_i)$ intersect, for all $i$.  But $\cap_i \varphi(w_i) = \varphi(e)$, and from the fact that geodesics in negatively curved spaces intersect at most once it is fairly easy to see that $\varphi(\gamma)$ intersects $\varphi(e)$, as desired.
\end{proof}


\begin{cor}\label{cor:vertex}
If a geodesic $\gamma$ passes through a vertex $v$, then $\varphi(\gamma)$ passes through $\varphi(v)$.
\end{cor}

\begin{proof}
Pick any apartment $A$ containing $\gamma$ and let $\{e_1, \ldots e_{2n}\}$ be the collection of edges in $A$ containing the vertex $v$. $\gamma$ passes through each of these so by Proposition \ref{prop:edgeint}, $\varphi(\gamma)$ hits all $\varphi(e_i)$. But $\varphi(v)$ is the intersection of all of these, so $\varphi(\gamma)$ intersects $\varphi(v)$.
\end{proof}

This implies a stronger version of Proposition \ref{geodwall}:


\begin{prop}
If the geodesic $\gamma$ and the wall $w$ intersect, then so do $\varphi(\gamma)$ and $\varphi(w)$.
\end{prop}

\begin{proof}
The only case not covered by Proposition \ref{geodwall} is when the intersection of $\gamma$ and $w$ is a vertex. This is handled by Corollary \ref{cor:vertex}.
\end{proof}

%
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\subsection{Good position} Having analyzed the possible intersections of a geodesic with a wall, we now
move to the more general situation: intersections of two arbitrary geodesics. We first identify a situation where it
is easy to ensure that the image geodesics still intersect.

\begin{defn}
Let $\gamma_1$ and $\gamma_2$ be two $g_0$-geodesics in $\tilde X$. We say that the pair $(\gamma_1,\gamma_2)$ is in \emph{good position} if there is some chamber $C$ in $\tilde X$ through which both $\gamma_i$ pass, and four distinct edges of $C$ such that $\gamma_1$ intersects the boundary of $C$ in edges $e_1$ and $e_3$, $\gamma_2$ intersects the boundary of $C$ in edges $e_2$ and $e_4$, and the cyclic order of the edges around the boundary of $C$ is $e_1, e_2, e_3, e_4$. (See Figure \ref{fig:good}).
\end{defn}


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\put(-110,-95){$e_4$}


\end{picture}

\begin{figure}[h]
\caption{Geodesics in good position.}\label{fig:good}
\end{figure}

\end{center}



It is clear that two geodesics in good position must intersect in a point. That yields the following:

 

\begin{prop}\label{prop:good intersect}
Let $\gamma_1$ and $\gamma_2$ be $g_0$-geodesics in good position. Then $\varphi(\gamma_1)$ and $\varphi(\gamma_2)$ intersect in a point. If $\gamma_1$ and $\gamma_2$ intersect in the interior of $C$ (and hence intersect transversally), then $\varphi(\gamma_1)$ and $\varphi(\gamma_2)$ intersect transversally.
\end{prop}


\begin{proof}
Using Proposition \ref{prop:edgeint}, $\varphi(\gamma_1)$ and $\varphi(\gamma_2)$ must be in good position, and hence must intersect in a point. The only cases where this intersection could fail to be transversal are if the intersection occurs at a vertex of $C$. But using Corollary \ref{cor:vertex} we see that this case does not occur for $\varphi(\gamma_1)$ and $\varphi(\gamma_2)$ if $\gamma_1\cap \gamma_2$ is not a vertex.
\end{proof}





%
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\subsection{Intersections of general geodesics}




Proposition \ref{prop:good intersect} takes care of a large class, but not all pairs of intersecting $g_0$-geodesics. The behavior of transversally intersecting geodesics which are not in good position is the focus of this section.


We remarked above in the proof of Proposition \ref{prop:good intersect} that the intersections detectable by 
good position are necessarily transversal if we avoid vertices. Throughout this section and below, we will restrict 
our attention to pairs of geodesics which intersect in non-vertex points. We note that the geodesics in such pairs 
are drawn from a $\varphi$-invariant (Corollary \ref{cor:vertex}), full measure set for any Liouville current.



\begin{defn}
Let $\gamma_1$ and $\gamma_2$ be a pair of $g_0$ geodesics intersecting transversally in an interior point 
of a chamber $C$. A \emph{window} for the pair $(\gamma_1, \gamma_2)$ is any union of chambers $C_i$ 
such that each $C_i$ intersects $\gamma_1$ and $\gamma_2$ in the interior of $C_i$. (See Figure \ref{fig:window}.)
\end{defn}



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\begin{figure}[h]
\caption{A window for $\gamma_1$ and $\gamma_2$. Note that it is not a good window (Defn \ref{defn:good window}) because the geodesics exit on the right through the same edge.}\label{fig:window}
\end{figure}

\end{center}

Note that if $\gamma_1$ and $\gamma_2$ are in good position, $C$ itself is the only window for 
$(\gamma_1, \gamma_2)$. In fact, for any window, both $\gamma_1$ and $\gamma_2$ must cross 
all the edges shared by more than one chamber in the window. Note that a window is a convex, 
two-dimensional subset of the building.


\begin{defn}\label{defn:good window}
Say $\gamma_1\cap \gamma_2$ is a non-vertex point. A window for $(\gamma_1, \gamma_2)$ is 
a \emph{good window} for the pair $(\gamma_1, \gamma_2)$ if $\gamma_1$ intersects its boundary 
in the edges $e_1, e_3$, $\gamma_2$ intersects its boundary in the edges $e_2, e_4$, and the cyclic 
ordering of these (distinct) edges around the boundary of the window is $e_1, e_2, e_3, e_4$.
\end{defn}


We will denote by $\varphi(W)$ the image in $(\tilde X,g_1)$ of a window in $(\tilde X,g_0)$ under 
the combinatorial isomorphism between the buildings.


\begin{lem}\label{lem:good window int}
If $W$ is a good window for $(\gamma_1, \gamma_2)$, then $\varphi(\gamma_1)$ and 
$\varphi(\gamma_2)$ intersect transversally in the good window $\varphi(W)$.
\end{lem}

\begin{proof}
Same as Proposition \ref{prop:good intersect}.
\end{proof}


For a geodesic $\gamma$ passing through a chamber $C$, let $\gamma|_C$ denote the geodesic 
segment $\gamma \cap C$.



\begin{defn}
A geodesic segment $\gamma'$ is an \emph{extension} of $\gamma|_C$ if $\gamma'\cap C =\gamma|_C$.
\end{defn}



\begin{defn}
Suppose $\gamma_1$ and $\gamma_2$ are $g_0$-geodesics intersecting in a non-vertex point of a 
chamber $C$. Let $\mathcal{GW}(\gamma_1, \gamma_2)$ be the set of all $W$ where $W$ is a good 
window for $(\gamma_1', \gamma_2')$ and where $\gamma_i'$ is an extension of $\gamma_i|_C$.
\end{defn}



\begin{lem}
The pair $(\gamma_1, \gamma_2)$ is already in good position if and only if 
$\mathcal{GW}(\gamma_1, \gamma_2) = \{C\}$.
\end{lem}

\begin{proof}
Clear.
\end{proof}



\begin{lem}
If $\gamma_1$ and $\gamma_2$ intersect in $C$, then every apartment through $C$ contains a good 
window for extensions of $(\gamma_1,\gamma_2)$, and $\mathcal{GW}(\gamma_1,\gamma_2)$ is finite.
\end{lem}

\begin{proof}
Let $A$ be an apartment through $C$ and $\gamma_i'$ the extensions of $\gamma_i|_C$ to $A$. Since 
they intersect transversely and the curvature is negative, they cannot stay at a bounded distance from one 
another, and thus the sequence of edges $\gamma_1'$ and $\gamma_2'$ pass through must differ 
eventually, in both directions. Thus they lie in a good window $W\subset A$. 


Further, the distance between $\gamma_1'$ and $\gamma_2'$ parametrized by arc length must diverge 
at least linearly, and so we can bound the diameter of the good window containing them. Since there are 
only finitely many restrictions of apartments to a given metric ball containing $C$, there are only finitely 
many good windows for the pair $(\gamma_1, \gamma_2)$.
\end{proof}



\begin{defn}
Let $DG_0(\tilde X)$ be the set of $g_0$-geodesic pairs $(\gamma_1,\gamma_2)$ intersecting 
transversally in $X$ in a non-vertex point. Let 

\[\mathcal{GW}(\tilde X) = \bigcup_{DG_0(\tilde X)} \mathcal{GW}(\gamma_1,\gamma_2).\]
\end{defn}

\begin{lem}
$\mathcal{GW}(\tilde X)$ is countable.
\end{lem}

\begin{proof}
We prove the Lemma for $\mathcal{GW}(C)$, the set of good windows for pairs intersecting transversally 
in $C$, then use the fact that there are countably many chambers in $\tilde X$.


There is a countable partition of $(0,\pi/2]$ into intervals $I_n$ (independent of choice of chamber or 
intersection point) such that if $\gamma_1$ and $\gamma_2$ intersect at angle $\theta \in I_n$ then 
any good window for $(\gamma_1, \gamma_2)$ contains at most $n$ chambers. As there are finitely 
many windows containing $C$ and containing $n$ chambers, $\mathcal{GW}(C)$ is a countable union 
of finite sets, proving the statement.
\end{proof}


\begin{defn}
Enumerate the windows in $\mathcal{GW}(\tilde X)$ as $\{W_i\}$. Then let 

\[ \widehat{DG}_0(\tilde X) = \bigcup_{i} \widehat{DG}_0(W_i) \]
where $\widehat{DG}_0(W_i)=\{(\gamma_1, \gamma_2): W_i \mbox{ is a good window for } (\gamma_1, \gamma_2) 
\mbox{ and } \gamma_1\cap \gamma_2 \neq\mbox{ a vertex}\}$.
\end{defn}


The following Lemma is easy to check.

\begin{lem}
For any pair $(\gamma_1, \gamma_2)$ of $g_0$-geodesics intersecting transversally in $\tilde X$, 
there is at least one $(\gamma_1',\gamma_2') \in \widehat{DG}_0(\tilde X)$ such that $\gamma_i'$ is an 
extension of $\gamma_i|_C$.
\end{lem}




%
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\subsection{$\widehat{DG}(X)$ and the intersection pairing}

We now show that $\widehat{DG}_0(\tilde X)$ descends to a well-defined, $\varphi$-invariant collection 
of intersecting geodesic pairs on $X$, and define our adjusted intersection pairing.

\begin{lem}
Let $\varphi$ send $g_0$-geodesics to $g_1$-geodesics as defined previously. Then

\[ \varphi(\widehat{DG}_0(\tilde X)) = \widehat{DG}_1(\tilde X) \]
where $\varphi((\gamma_1, \gamma_2))=(\varphi(\gamma_1), \varphi(\gamma_2)).$
\end{lem}

\begin{proof}
If $(\gamma_1, \gamma_2) \in \widehat{DG}_0(\tilde X)$ with $W$ a good window for the pair, then by 
Lemma \ref{lem:good window int}, $\varphi(W)$ is a good window for $(\varphi(\gamma_1),\varphi(\gamma_2))$. 
Therefore $\varphi(\gamma_1)$ and $\varphi(\gamma_2)$ intersect in some chamber belonging to $\varphi(W)$ 
(and not in a vertex, by Corollary \ref{cor:vertex}). Then 
$(\varphi(\gamma_1), \varphi(\gamma_2)) \in \widehat{DG}_1(\tilde X)$. Thus 
$\varphi(\widehat{DG}_0(\tilde X)) \subseteq \widehat{DG}_1(\tilde X)$.
Reversing the roles of $g_0$ and $g_1$ gives the other inclusion and completes the proof.
\end{proof}



Given this lemma, we may without ambiguity write $\widehat{DG}(\tilde X)$ and suppress the dependence on 
the metric, unless we specifically want to view a pair of geodesics in terms of one metric or the other.







%If $(\gamma_1,\gamma_2) \in \widehat{DG}(\tilde X)$ have good window $W$, then $\phi(\gamma_1,\gamma_2)$ have good window $\phi(W)$. But we do not know whether the intersection between $\phi(\gamma_1)$ and $\phi(\gamma_2)$ takes place in $\phi(C)$ where $C$ is the chamber where $\gamma_1$ and $\gamma_2$ intersect. Rewrite

%\[ \widehat{DG}(\tilde X) = \bigcup_{i} \widehat{DG}(W_i) \]
%where $\widehat{DG}(W_i)$ consists of all $(\gamma_1,\gamma_2)$ such that $W_i$ is a good window for $(\gamma_1,\gamma_2)$, thereby removing the reference to the chamber $C$.  It is clear from the definition of a good window and Proposition \ref{prop:edgeint} that this is a disjoint union.



\begin{lem}\label{lem:gamma inv}
Under the obvious action, $\widehat{DG}(\tilde X)$ is $\Gamma$-invariant and $\varphi:\widehat{DG}_0(\tilde X)\to \widehat{DG}_1(\tilde X)$ is $\Gamma$-equivariant.
\end{lem}

\begin{proof}
This is clear, as $\Gamma$ acts by isometries, which also preserve the combinatorial structure of the building.
\end{proof}


The first part of Lemma \ref{lem:gamma inv} justifies this definition:

\begin{defn}
Let $\widehat{DG}(X) := \widehat{DG}(\tilde X)/\Gamma$.
\end{defn}

Lemma \ref{lem:gamma inv} then has the following immediate consequence:

\begin{lem}
The map $\varphi$ descends to a map $\varphi:\widehat{DG}_0(X) \to \widehat{DG}_1(X)$.
\end{lem}


\begin{lem}
%The image of the collection $\{W_i\}$ of good windows descends under $\tilde X\to X$ to a countable collection $\{U_i\}$ so that
Let $\{U_i\}$ denote the countable collection of $\Gamma$-orbits in the collection $\{W_i\}$, and for each 
$U_i = \Gamma \cdot W_i$, set $\widehat{DG}(U_i):= \Gamma\cdot \widehat{DG}(W_i)$. Then we have a
disjoint union
\[ \widehat{DG}(X) = \bigcup_i \widehat{DG}(U_i).\]
%where $\widehat{DG}(U_i)$ is the image under $\tilde X\to X$ of the $\Gamma$-translates of some $\widehat{DG}(W_i)$. This union is disjoint.
\end{lem}

\begin{proof}
This is again easy to check as $\Gamma$ acts by isometries and preserves the combinatorial structure of the building. 
\end{proof}



We now define the `adjusted' intersection number. Let $e$ be an edge in $X$. Recall that $q(e)$ is the integer such 
that there are $q(e)$ chambers containing the edge $e$. As the building is thick, $q(e)\geq 3$.


\begin{defn}
Let $W$ be a window and let $\{e_1, \ldots e_n\}$ be the set of edges shared by consecutive pairs of 
chambers in $W$. The \emph{weight} of $W$ is defined as

\[ \varpi(W) = \prod_{i=1}^n (q(e_i)-1). \] 
\end{defn}




\begin{defn}
Let $\mu$ and $\lambda$ be geodesic currents on $X$. We define their \emph{adjusted intersection number} as

\[ \hat i(\mu,\lambda) = \sum_i \varpi(U_i)(\mu \times \lambda)(\widehat{DG}(U_i)). \]
\end{defn}




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\subsection{Intersection number calculations}


To justify this adjusted notion of intersection number, we want to calculate the following three standard types of pairings:

\begin{itemize}
	\item $\hat i(\alpha, \beta)$ where $\alpha$ and $\beta$ are the geodesic currents defined by Dirac unit masses supported on the closed geodesics $\alpha$ and $\beta$.
	\item $\hat i(L_g,\beta)$ where $L_g$ is the Liouville current.
	\item $\hat i(L_g,L_g)$.
\end{itemize}




\begin{prop}\label{prop:closed int} When $\alpha, \beta$ are a pair of geodesic currents supported on closed geodesics, we have
\[ \hat i(\alpha,\beta) = \sum_j \mathfrak{p}_j \]
where $j$ enumerates the intersections of the geodesics $\alpha$ and $\beta$ in $X$ and $\mathfrak{p}_j = \varpi(U_j)$ where $U_j$ is the good window witnessing the $j^{th}$ intersection of these geodesics if such a window exists, and $\mathfrak{p}_j=0$ if no good window witnessing the $j^{th}$ intersection exists (or if the intersection occurs at a vertex).
\end{prop}


\begin{cor}\label{cor:good int number}
If all intersections between $\alpha$ and $\beta$ are in good position and not at vertices, then $\hat i(\alpha,\beta)$ is the number of intersections.
\end{cor}

\begin{proof}[Proof of Cor \ref{cor:good int number}]
In this case, all $\mathfrak{p}_j=1$.
\end{proof}


\begin{rem}
In the case of Corollary \ref{cor:good int number}, we recover the intersection number of the two closed geodesics, 
exactly as in the surface case.
\end{rem}



\begin{proof}[Proof of Prop \ref{prop:closed int}]
By our definition of the intersection pairing, all contributions to $\hat i(\alpha,\beta)$ come from measuring $\widehat{DG}(U_i)$. Since the currents $\alpha$ and $\beta$ are supported on the single geodesics $\alpha$ and $\beta$, the contribution from $\widehat{DG}(U_i)$ is non-zero if and only if $U_i$ is a good window for $(\alpha,\beta)$. Thus, there is a contribution of $\varpi(U_i)$ to $\hat i(\alpha,\beta)$ precisely for each intersection witnessed by the good window $U_i$, and those intersections which are not witnessed by any good window contribute nothing.
\end{proof}



\begin{prop}\label{prop:length}
For any closed geodesic which is not tangent to a wall,
\[ \hat i(L_g,\beta) = 2 \cdot \text{Length}_g(\beta). \]
\end{prop}


\begin{proof}
As in Proposition \ref{prop:closed int}, the only contributions to $\hat i(L_g,\beta)$ will come from windows $U_i$ which are good for $(\gamma,\beta)$ for some geodesic $\gamma$. For such windows, $(L_g\times \beta)(\widehat{DG}(U_i)) = L_g(A_i)$ where $A_i$ consists of the geodesics $\gamma$ which pair with $\beta$ so that $U_i$ is a good window for $(\gamma, \beta)$. From the definition of $L_g$ we have that 

\[ L_g(A_i) = \frac{1}{\varpi(U_i)} \int_{A_i} \cos \theta d\theta dp \]
where the integral is over the intersections of the geodesics in $A_i$ with $\beta$, and $\theta$ and $p$ are the 
coordinates for these intersections with respect to the metric $g$. Therefore,

\[ \varpi(U_i)(L_g \times \beta)(\widehat{DG}(U_i)) = \int_{A_i} \cos\theta d\theta dp. \]


Since every vector with basepoint on $\beta$ and making a nonzero angle with $\beta$ generates exactly one geodesic $\gamma$ such that $(\gamma,\beta)$ lie in a good window $U_i$, summing the expression above over all $i$ gives

\[\hat  i(L_g,\beta) =  \int_{t\in \beta, \theta \in(-\frac{\pi}{2},\frac{\pi}{2})} \cos\theta d\theta dt = 2\cdot \text{Length}_g(\beta).\] 
\end{proof}


\begin{rem}
This result also reproduces the analogous result for surfaces. For geodesics along walls, one obtains $\hat i_g(L_g,\beta) = K\cdot \text{Length}_g(\beta)$, where $K$ is the number of chambers adjacent to any edge in the wall.
\end{rem}



\begin{prop}\label{prop:vol}
\[ \hat i(L_g,L_g) = \frac{1}{4\pi}Vol_g(X).\]
\end{prop}



\begin{proof}
We compute chamber by chamber, noting that 

\[ Vol_g(X) = \sum_C Vol_g(C). \]

If we denote by $S^1C$ the unit tangent vectors to $C$ based at edge points and pointing into $C$, then

\[ Vol_g(C) = \frac{1}{2\pi} \int_{v\in S^1C} l_g(v) d\mu \]
where $d\mu = \cos\theta d\theta dp$ is a measure on $S^1C$ and $l_g(v)$ is the length of the geodesic segment in $C$ generated by $v$. Call this segment $\gamma_v$.



Now we also have

\[ l_g(v)=\frac{1}{2} \int_{A_v} 1 d\mu \]
where $d\mu = \cos\theta d\theta dp$ is the same measure but for tangent vectors along $\gamma_v$, and $A_v$ is 
the set of geodesics intersecting $\gamma_v$ transversally in $C$.

Consider the pairs $(\gamma_v, \eta)$ with $\eta$ a geodesic intersecting $\gamma_v$ transversally in $C$. Not all of these are in good position, so we replace them with 
\[\mathcal{G}_i(v) = \{(\gamma_v', \eta): U_i \mbox{ is a good window for }(\gamma_v',\eta) \mbox{ where } \gamma_v' \mbox{ is an extension of } \gamma_v\}.\]

Then, just as in the proof of Proposition \ref{prop:length}, 
\[ l_g(v) = \frac{1}{2}\sum_i\int_{\mathcal{G}_i(v)} \varpi(U_i) \frac{d\delta_{\gamma_v'}}{\varpi(U_i)}dL_g(\eta) \]
where $\delta_{\gamma_v'}$ is the Dirac mass on $\gamma_v'$. We must divide by the factor $\varpi(U_i)$ to correct for the fact that there are multiple extensions $\gamma_v'$ of $\gamma_v$. Weighting by $\frac{1}{\varpi(U_i)}$ precisely cancels out this error.


We now have
\[ Vol_g(C) = \frac{1}{4\pi} \int_{v\in S^1C} \sum_i \int_{\mathcal{G}_i(v)} \varpi(U_i) \frac{d\delta_{\gamma_v'}}{\varpi(U_i)}dL_g(\eta) d\mu. \]
From the description of $L_g$, $d\delta_{\gamma_v'}d\mu = \varpi(U_i) dL_g$, so
\[ Vol_g(C) = \frac{1}{4\pi} \sum_i \int_{\mathcal{G}_i(C)} \varpi(U_i) dL_g(\eta) dL_g(\gamma')\]
where the sets $\mathcal{G}_i(C)$ are defined by
\[ \mathcal{G}_i(C) = \{(\gamma',\eta):  U_i \mbox{ is a good window for }(\gamma',\eta) \mbox{ where } \gamma' \mbox{ intersects } \eta \mbox{ transversally in } C\}. \]

Summing this over all $C$, we have 
\[ Vol_g(X) = \frac{1}{4\pi} \sum_{i} \varpi(U_i)(L_g\times L_g)(\widehat{DG}(U_i)) \]
which is just
\[ Vol_g(X) = \frac{1}{4\pi} \hat i(L_g,L_g), \]
completing the proof.
\end{proof}


\begin{rem}
Again, this reproduces the analogous result for surfaces.
\end{rem}



Another key property of the intersection pairing is finiteness, some cases of which are proven above. We will also need


\begin{prop}\label{prop:finiteness}
\[ \hat i(L_{g_0},L_{g_1}) < \infty. \]
\end{prop}


\begin{proof}
Let $G(U_i)$ be the set of all geodesics which appear as one of the two geodesics in any element of 
$\widehat{DG}(U_i)$ (recall such elements are pairs of geodesics). Fix, for each good window $U_i$, a chamber $C$ contained in $U_i$ and let $A_i$ be the set of geodesics whose restriction to that chamber agrees with the restriction of a geodesic in $G(U_i)$. As noted above, $L_g(A_i) = \varpi(U_i)L_g(G(U_i))$. Denote by $|U_i|$ the number of chambers in $U_i$.

We thus have


\begin{align*}
	\hat i(L_{g_0},L_{g_1})  & = \sum_i\varpi(U_i)(L_{g_0}\times L_{g_1})(\widehat{DG}(U_i))  \\
		&\leq \sum_i\varpi(U_i)L_{g_0}(G(U_i))L_{g_1}(G(U_i))  \\
		& = \sum_i \frac{1}{\varpi(U_i)}L_{g_0}(A_i)L_{g_1}(A_i)  \\
		& = \sum_k \sum_{|U_i|=k} \frac{1}{\varpi(U_i)}L_{g_0}(A_i)L_{g_1}(A_i)
\end{align*}

Since geodesics diverge exponentially fast, it is easy to see that there is an exponentially decaying function $f(k)$ such that $L_{g_i}(A_i)\leq f(k)$ where $|U_i|=k$ for all $i$. We then continue:

\begin{align*}
	& \leq \sum_k f(k)^2 \sum_{|U_i|=k} \frac{1}{\varpi(U_i)} \\
	& \leq \sum_k f(k)^2 \sum_{C\subset X/\Gamma} \ \sum_{|U_i|=k, \ U_i\supset C} \frac{1}{\varpi(U_i)}.
\end{align*}


It is easy to check that 

\[ \sum_{|U_i|=k, \ U_i\supset C} \frac{1}{\varpi(U_i)} \leq 1 \]
so we have in total that 

\[ \hat i(L_{g_0},l_{g_1}) \leq \#C \sum_k f(k)^2. \]

Since $f(k)$ decays to zero exponentially fast, the sum converges, finishing the proof.


\end{proof}



We will also need the following lemma. The invariance under $\varphi$ of $\hat i(-,-)$ it expresses is the reason we had to re-define the intersection pairing.

\begin{lem}
If $\varphi: G_0(X) \to G_1(X)$ is the identification of $g_0$- and $g_1$-geodesics and $\mu, \lambda$ are two currents on $G_0(X)$, then

\[ \hat i(\varphi_*\mu, \varphi_*\lambda) = \hat i(\mu,\lambda). \] 
\end{lem}

\begin{proof}
$\hat i(\varphi_*\mu,\varphi_*\lambda) = \sum_i \varpi(U_i)(\varphi_*\mu \times \varphi_*\lambda)(\widehat{DG}(U_i))$. But $\widehat{DG}(U_i)$ is invariant under $\varphi$, as is the combinatorial factor $\varpi(U_i)$. The result follows.
\end{proof}








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\section{Continuity of $\hat i(-,-)$}\label{sec:cont}



The final, and crucial, property of the intersection pairing is some form of continuity. In the setting of geodesic currents on compact surfaces, the intersection pairing $i(-,-)$ is continuous in both entries. In the Fuchsian building setting, given the difficulties involving transverse intersection and the new definition we adopted, it is not clear whether $\hat i(-,-)$ is always continuous. Nevertheless, we can establish the following weaker result, which is sufficient for our purposes:



\begin{prop}\label{prop:hard continuity}
Let $g_0, g_1$ be any metrics in $\mathcal{M}_{neg}^v$. Say $\alpha_n$ are supported on closed geodesics such that $\alpha_n \to L_{g_0}$. Then

\[ \hat i(\alpha_n, L_{g_1}) \to \hat i(L_{g_0},L_{g_1}). \]
\end{prop}

\begin{proof}
By definition,
\[ \hat i(\alpha_n,L_{g_1}) = \sum_i \varpi(U_i)(\alpha_n\times L_{g_1})(\widehat{DG}(U_i)). \]
Note that each $\widehat{DG}(U_i)$, as a subset of $\mathcal{G}(X)\times \mathcal{G}(X)$, is compact. So the fact that $\alpha_n\times L_{g_1}$ weak-* converges to $L_{g_0}\times L_{g_1}$ implies that for all $i$,
\[ (\alpha_n \times L_{g_i})(\widehat{DG}(U_i)) \to (L_{g_0}\times L_{g_1})(\widehat{DG}(U_i)). \]
To deal with the (infinite) sum in the expression for $\hat i(-,-)$, we will prove the following claim:

\vspace{10pt}
\noindent\textbf{Claim:} Given $\epsilon>0$, there exists some $I_1$ such that for all $n$,
\[ \sum_{i>I_1} \varpi(U_i)(\alpha_n\times L_{g_1})(\widehat{DG}(U_i)) <\epsilon. \]
\vspace{10pt}

Once this claim is established, the proof of the proposition is as follows. Let $\epsilon>0$ be given. Let $I_1$ be as provided by the Claim. Further, since $\hat i(L_{g_0},L_{g_1})<\infty$ (Proposition \ref{prop:finiteness}), there exists some $I_2$ such that
\[ \sum_{i>I_2} \varpi(U_i)(L_{g_0}\times L_{g_1})(\widehat{DG}(U_i)) <\epsilon, \]
since the sum on $i$ converges.  Let $I=\max\{I_1, I_2\}$. Pick $N$ so large that $n>N$ implies that 
\[ \big|\sum_{i=1}^I \varpi(U_i)(\alpha_n\times L_{g_1})(\widehat{DG}(U_i)) - \sum_{i=1}^I \varpi(U_i)(L_{g_0}\times L_{g_1})(\widehat{DG}(U_i)) \big| <\epsilon, \]
using the weak-* convergence of $\alpha_n$ to $L_{g_0}$ as noted above. Putting these three inequalities together, we get that for $n>N$,

\[ \big|\hat i(\alpha_n,L_{g_1}) - \hat i(\alpha_n,L_{g_0})\big| < 3\epsilon, \]
finishing the proof.

\begin{proof}[Proof of Claim:]

Let $G(U_i)$ be the set of all geodesics involved in a pair for which $U_i$ is a good window. Note that $G(U_i)$ consists of all geodesics which traverse all the chambers making up $U_i$ consecutively. Fix any chamber $C$ in $U_i$. Let $A_i$ be the set of $g_1$-geodesics which agree with the restriction to $C$ of the geodesics in $G(U_i)$. From the description of $L_{g_1}$, we see that $L_{g_1}(G(U_i)) = \frac{1}{\varpi(U_i)}L_{g_1}(A_i)$.


Now as the number of chambers in $U_i$ increases, we see that the restriction of the geodesics in $G(U_i)$ to $C$ form a smaller and smaller set. In fact, since the curvature is bounded strictly away from zero and below zero, and geodesics in negative curvature diverge exponentially fast, it must be the case that $L_{g_1}(A_i)$ tends to zero exponentially fast as the number of chambers in $U_i$ increases (not necessarily as $i$ increases). Let $f(k)$ be a function decaying to zero exponentially fast so that if $U_i$ has $k$ chambers, $L_{g_1}(U_i)< f(k)$.


Note that $L_{g_0}(G(X))$ is by definition finite, and as $\alpha_n \to L_{g_0}$, we see that $\alpha_n(G(U_i))\leq \alpha_n(G(X))$ is bounded, say by $K$. 


Recall that $\alpha_n$ is a Dirac measure supported on a closed geodesic; $\alpha_n(G(U_i))$ is zero if the closed geodesic $\alpha_n$ does not belong to $G(U_i)$. For any window with $k$ chambers, the value of $\alpha_n(G(U_i))$ is (proportional to) the number of times $\alpha_n$ runs through $U_i$. Putting an arbitrary orientation on the closed geodesic $\alpha_n$, we see that for each time $\alpha_n$ runs through $U_i$, there is a unique extension $U_i'$ of $U_i$ by one chamber in the forward direction of $\alpha_n$ so that that $\alpha_n$ runs through the extended window. From this, by induction, we see that

\begin{equation}\label{eqn:size}
	 \sum_{U_i \mbox{ of size }1} \alpha_n(G(U_i)) = \sum_{U_i \mbox{ of size a fixed }k>1} \alpha_n(G(U_i)). 
\end{equation}

Combining all this, we get

\[ \sum_i \varpi(U_i)(\alpha_n\times L_{g_1})(\widehat{DG}(U_i)) \leq \sum_i\varpi(U_i)\alpha_n(G(U_i))L_{g_1}(G(U_i)) \]
\[ = \sum_i \alpha_n(G(U_i))L_{g_1}(A_i)\]
\[ = \sum_k \sum_{U_i \mbox{ containing } k \mbox{ chambers}} \alpha_n(G(U_i))L_{g_1}(A_i)\]

Letting $A_i$ be the segments in the first chamber $C$ of $U_i$ corresponding to $G(U_i)$ and using equation \eqref{eqn:size}, 

\[ = \sum_k \sum_C \alpha_n(G(C)) L_{g_1}(A_i). \]

Note that $k$ indexes the number of chambers involved in $U_i$ inducing $A_i$. Using our bounding function $f(k)$,

\[ < \sum_k \sum_C \alpha_n(G(C)) f(k). \]
\[ = \sum_k f(k) \sum_C \alpha_n(G(C)). \]
\[ \leq \sum_k f(k) (\# \mbox{chambers}) \alpha_n(G(X)) \]
\[ \leq \sum_k f(k) (\# \mbox{chambers}) K<\infty\]


Since the sum converges, there must exist some $N_\epsilon$ such that 

\[ \sum_{k>N_\epsilon} f(k) (\# \mbox{chambers}) K< \epsilon. \]


Letting $I_i$ be the number of windows in $X$ of length no more than $N_\epsilon$, we see that 

\[ \sum_{i>I_1} \varpi(U_i)(\alpha_n\times L_{g_1})(\widehat{DG}(U_i)) < \sum_{k>N_\epsilon} f(k)(\#\mbox{chambers})K <\epsilon \]
with $I_1$ independent of $n$ as desired.
\end{proof}
\end{proof}


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\section{Marked length spectrum and the volume function}\label{vol_conject}



To prove Theorem \ref{thm:MLS-determines-volume}, we move from the length inequality to the volume inequality via the intersection pairing. Our proof follows that in \cite{croke-dairbekov}.


\begin{proof}[Proof of Theorem \ref{thm:MLS-determines-volume}]
For any closed geodesic $\alpha_n$ not along a wall (using Corollary \ref{wall-corollary}), 
\[ \hat i(\alpha_n,L_{g_0}) = 2 \cdot \text{Length}_{g_0}(\alpha_n) \leq 2 \cdot \text{Length}_{g_1}(\alpha_n) = \hat i(\alpha_n,L_{g_1}). \]


By Proposition \ref{density} there is a sequence $\alpha_n\to L_{g_0}$. Since $L_{g_0}$ gives zero measure to geodesics along walls, we can assume that no $\alpha_n$ is along a wall. Applying Proposition \ref{prop:hard continuity} twice,
\[ \hat i(L_{g_0},L_{g_0}) \leq \hat i(L_{g_0},L_{g_1}). \]
Similarly, for any closed geodesic $\beta_n$ not along a wall,
\[ \hat i(L_{g_0},\beta_n) = 2\cdot \text{Length}_{g_0}(\beta_n) \leq 2\cdot \text{Length}_{g_1}(\beta_n) = \hat i(L_{g_1},\beta_n). \]
Letting $\beta_n\to L_{g_1}$ and applying the same continuity result,
\[ \hat i(L_{g_0},L_{g_1}) \leq \hat i(L_{g_1},L_{g_1}). \]
Combining these inequalities, we have

\[ \frac{1}{4\pi} Vol_{g_0}(X) = \hat i(L_{g_0},L_{g_0}) \leq \hat i(L_{g_0},L_{g_1}) \leq \hat i(L_{g_1},L_{g_1}) =  \frac{1}{4\pi} Vol_{g_1}(X)\]
as desired.
\end{proof}



\begin{rem}
We expect that Theorem \ref{thm:MLS-determines-volume} holds for any pair of metrics in $\mathcal{M}_\leq(X)$, without the non-singular vertices condition. Our arguments would prove this if one could show that, for a sequence $\alpha_n\to L_g$, the corresponding $g'$-geodesics spend proportion $p_n \to 0$ of the time in the 1-skeleton of $(X,g')$. Without such a result, we can only ensure that $\hat i(\alpha_n,L_g)\geq 2\cdot \text{Length}_g(\alpha_n)$ (see the Remark after Proposition \ref{prop:length}). This breaks the arguments for $\hat i(L_{g_0},L_{g_0})\leq \hat i(L_{g_0},L_{g_1})$ and $\hat i(L_{g_0},L_{g_1})\leq \hat i(L_{g_1},L_{g_1})$ above. This is the only place in our argument for Theorem \ref{thm:MLS-determines-volume} where the non-singular vertices assumption is used.
\end{rem}





\begin{thebibliography}{XXX}

\bibitem[ABr]{brown} P. Abramenko and K. Brown. \textit{Buildings: Theory and Applications}. Springer, 2008.

\bibitem[AB]{alperin-bass} R. Alperin and H. Bass. \textit{Length functions of group actions on $\lambda$-trees}. 
In ``Combinatorial group theory and topology (Alta, Utah, 1984)'', 265--378. Ann. of Math. Stud. \textbf{111}, 
Princeton Univ. Press, Princeton, NJ, 1987.

\bibitem[BB]{ball_brin} W. Ballmann and M. Brin. \textit{Orbihedra of nonpositive curvature}. Publ. Math. IHES \textbf{82} (1995), 169--209.

%\bibitem[Bon1]{bonahon_ends} F. Bonahon. \textit{Bouts des vari\'et\'es hyperboliques en dimension $3$}. Annals Math. \textbf{124} (1986), 71--158.
\bibitem[BL]{bankovic-leininger} A. Bankovi\'c and C. Leininger. \textit{Marked length spectral rigidity for flat metrics}. 
Preprint available on the arXiv:1504.01159

\bibitem[Bon1]{bonahon_geometry} F. Bonahon. \textit{The geometry of Teichm\"uller space via geodesic currents}. Invent. Math. \textbf{92} (1988), 139--162.

\bibitem[Bon2]{bonahon_negative} F. Bonahon. \textit{Geodesic currents on negatively curved groups}. In ``Arboreal group theory'', 143--168.
MSRI Publications \textbf{19}, Springer-Verlag, 1991. 

\bibitem[Bou1]{bourdon} M. Bourdon. \textit{Sur le birapport au bord des CAT(-1)-espaces}. Publ. Math. IHES. \textbf{83} (1996), 95--104.

\bibitem[Bou2]{bourdon-3} M. Bourdon. \textit{Immeubles hyperboliques, dimension conforme, et rigidit\'e de Mostow}. Geom. Funct. Anal. \textbf{7} (1997), 245--268.

\bibitem[Bou3]{Bou} M. Bourdon. \textit{Sur les immeubles Fuchsiens et leur type de quasi-isom\'etrie}. 
Ergodic Theory Dynam. Systems \textbf{20} (2000), 343--364.

\bibitem[BP]{BP} M. Bourdon and H. Pajot. \textit{Rigidity of quasi-isometries for some hyperbolic buildings.} Comment. Math. Helv. \textbf{75} (2000), 701--736.

\bibitem[Co]{constantine} D. Constantine. \textit{Marked length spectrum rigidity in nonpositive curvature with singularities}.
Preprint available on the arXiv:1507.04970

\bibitem[CL]{CL} D. Constantine and J.-F. Lafont. \textit{Marked length rigidity for one dimensional spaces}. 
Preprint available on the arXiv:1209.3709

\bibitem[CFS]{CFS} I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai. \textit{Ergodic Theory}, Springer-Verlag, 1982.

\bibitem[Cr]{croke} C. B. Croke. \textit{Rigidity for surfaces of nonpositive curvature}. 
Comment. Math. Helv. \textbf{65} (1990), 150--169.

\bibitem[CrD]{croke-dairbekov} C. B. Croke and N. S. Dairbekov.
\textit{Lengths and volumes in Riemannian manifolds}. 
Duke Math. J. \textbf{125} (2004), 1--14.

\bibitem[CM]{culler-morgan} M. Culler and J. W. Morgan. \textit{Group actions on $\mathbb R$-trees}. 
Proc. London Math. Soc. (3) \textbf{55} (1987), 571--604.

\bibitem[DK]{dalbo-kim} F. Dal'Bo and I. Kim. \textit{Marked length rigidity for symmetric spaces}. 
Comment. Math. Helv. \textbf{77} (2002), 399--407.

\bibitem[DMV]{DMV} G. Daskalopoulos, C. Mese, and A. Vdovina. \textit{Superrigidity of hyperbolic buildings}. Geom. Funct. Anal. \textbf{21} (2011), 905--919.

\bibitem[Fa]{Fa} H.-R. Fana\"i.
\textit{Comparaison des volumes des vari\'et\'es riemanniennes}.
C. R. Math. Acad. Sci. Paris \textbf{339} (2004), 199--201. 

\bibitem[FH]{FH} W. Feit and G. Higman, \textit{The nonexistence of certain generalized polygons}. 
J. Algebra \textbf{1} (1964), 114--131.

\bibitem[Ha]{ham} U. Hamenst\"adt. \textit{Entropy-rigidity of locally symmetric spaces of negative curvature}. 
Ann. of Math. (2) \textbf{131} (1990), 35--51.

\bibitem[HP]{hersonsky-paulin} S. Hersonsky and F. Paulin. {\it On the rigidity of discrete isometry groups of negatively curved spaces}. 
Comment. Math. Helv. \textbf{72} (1997), 349--388. 

\bibitem[LL]{LL} F. Ledrappier and S.-H. Lim. \textit{Volume entropy for hyperbolic buildings}. J. Modern Dyn. \textbf{4} (2010), 139--165. 

\bibitem[Ot1]{otal} J.-P. Otal. \textit{Le spectre marqu\'e des longuers des surfaces \`a courbure n\'egative}. 
Ann. of Math. (2) \textbf{131} (1990), 151--162.

\bibitem[Ot2]{otal-2} J.-P. Otal. \textit{Sur la g\'eom\'etrie symplectique de l'espace des g\'eod\'esiques d'une vari\'et\'e \`a courbure n\'egative}. Revista Matematica Iberoamericana \textbf{8} (1992), 441--456.

\bibitem[PT]{payne-thas} S. E. Payne and J. A. Thas.
\textit{Finite generalized quadrangles}. 
European Mathematical Society (EMS), 2009. 


%\bibitem[S]{S} J. \'Swi\c{a}tkowski, \textit{Trivalent polygonal complexes of nonpositive curvature and Platonic symmetry.} Geom Dedicata. \textbf{70} (1998), 
%pgs. 87--110.

%\bibitem[Th]{Th} A. Thomas, \textit{Lattices in hyperbolic buildings}. To appear in the volume ``Geometry, topology, and dynamics in negative curvature - Proceedings'', London Math. Soc., Lecture Notes series. Preprint available on the 
%arXiv:1204.0287

\bibitem[S]{sun} Z. Sun. \textit{Marked length spectra and areas of non-positively curved cone metrics}. 
Geom. Dedicata \textbf{178} (2015), 189--194.

\bibitem[vM]{van-Maldeghem} H. van Maldeghem. \textit{Generalized polygons}. 
Birkh\"auser Verlag, 1998. 

\bibitem[We]{Weiss-structure} R. M. Weiss. \textit{The structure of spherical buildings}.
Princeton University Press, 2003.

\bibitem[Wi]{W} D.T. Wise, \textit{A flat plane that is not the limit of periodic flat planes}. 
Algebr. Geom. Topol. \textbf{3} (2003), pgs. 147--154. 

\bibitem[X]{X} X. Xie, \textit{Quasi-isometric rigidity of {F}uchsian buildings}. 
Topology \textbf{45} (2006), pgs. 101--169. 


\end{thebibliography}

\end{document}

1503.01321.extracted_bib
[ABr]{brown P. Abramenko and K. Brown. Buildings: Theory and Applications. Springer, 2008.
1503.01321.extracted_bib
[AB]{alperin-bass R. Alperin and H. Bass. Length functions of group actions on $\lambda$-trees. In ``Combinatorial group theory and topology (Alta, Utah, 1984)'', 265--378. Ann. of Math. Stud. 111, Princeton Univ. Press, Princeton, NJ, 1987.
1503.01321.extracted_bib
[BB]{ball_brin W. Ballmann and M. Brin. Orbihedra of nonpositive curvature. Publ. Math. IHES 82 (1995), 169--209. %
1503.01321.extracted_bib
[Bon1]{bonahon_ends F. Bonahon. Bouts des vari\'et\'es hyperboliques en dimension $3$. Annals Math. 124 (1986), 71--158.
1503.01321.extracted_bib
[BL]{bankovic-leininger A. Bankovi\'c and C. Leininger. Marked length spectral rigidity for flat metrics. Preprint available on the arXiv:1504.01159
1503.01321.extracted_bib
[Bon1]{bonahon_geometry F. Bonahon. The geometry of Teichm\"uller space via geodesic currents. Invent. Math. 92 (1988), 139--162.
1503.01321.extracted_bib
[Bon2]{bonahon_negative F. Bonahon. Geodesic currents on negatively curved groups. In ``Arboreal group theory'', 143--168. MSRI Publications 19, Springer-Verlag, 1991.
1503.01321.extracted_bib
[Bou1]{bourdon M. Bourdon. Sur le birapport au bord des CAT(-1)-espaces. Publ. Math. IHES. 83 (1996), 95--104.
1503.01321.extracted_bib
[Bou2]{bourdon-3 M. Bourdon. Immeubles hyperboliques, dimension conforme, et rigidit\'e de Mostow. Geom. Funct. Anal. 7 (1997), 245--268.
1503.01321.extracted_bib
[Bou3]{Bou M. Bourdon. Sur les immeubles Fuchsiens et leur type de quasi-isom\'etrie. Ergodic Theory Dynam. Systems 20 (2000), 343--364.
1503.01321.extracted_bib
[BP]{BP M. Bourdon and H. Pajot. Rigidity of quasi-isometries for some hyperbolic buildings. Comment. Math. Helv. 75 (2000), 701--736.
1503.01321.extracted_bib
[Co]{constantine D. Constantine. Marked length spectrum rigidity in nonpositive curvature with singularities. Preprint available on the arXiv:1507.04970
1503.01321.extracted_bib
[CL]{CL D. Constantine and J.-F. Lafont. Marked length rigidity for one dimensional spaces. Preprint available on the arXiv:1209.3709
1503.01321.extracted_bib
[CFS]{CFS I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai. Ergodic Theory, Springer-Verlag, 1982.
1503.01321.extracted_bib
[Cr]{croke C. B. Croke. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65 (1990), 150--169.
1503.01321.extracted_bib
[CrD]{croke-dairbekov C. B. Croke and N. S. Dairbekov. Lengths and volumes in Riemannian manifolds. Duke Math. J. 125 (2004), 1--14.
1503.01321.extracted_bib
[CM]{culler-morgan M. Culler and J. W. Morgan. Group actions on $\mathbb R$-trees. Proc. London Math. Soc. (3) 55 (1987), 571--604.
1503.01321.extracted_bib
[DK]{dalbo-kim F. Dal'Bo and I. Kim. Marked length rigidity for symmetric spaces. Comment. Math. Helv. 77 (2002), 399--407.
1503.01321.extracted_bib
[DMV]{DMV G. Daskalopoulos, C. Mese, and A. Vdovina. Superrigidity of hyperbolic buildings. Geom. Funct. Anal. 21 (2011), 905--919.
1503.01321.extracted_bib
[Fa]{Fa H.-R. Fana\"i. Comparaison des volumes des vari\'et\'es riemanniennes. C. R. Math. Acad. Sci. Paris 339 (2004), 199--201.
1503.01321.extracted_bib
[FH]{FH W. Feit and G. Higman, The nonexistence of certain generalized polygons. J. Algebra 1 (1964), 114--131.
1503.01321.extracted_bib
[Ha]{ham U. Hamenst\"adt. Entropy-rigidity of locally symmetric spaces of negative curvature. Ann. of Math. (2) 131 (1990), 35--51.
1503.01321.extracted_bib
[HP]{hersonsky-paulin S. Hersonsky and F. Paulin. {On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv. 72 (1997), 349--388.
1503.01321.extracted_bib
[LL]{LL F. Ledrappier and S.-H. Lim. Volume entropy for hyperbolic buildings. J. Modern Dyn. 4 (2010), 139--165.
1503.01321.extracted_bib
[Ot1]{otal J.-P. Otal. Le spectre marqu\'e des longuers des surfaces \`a courbure n\'egative. Ann. of Math. (2) 131 (1990), 151--162.
1503.01321.extracted_bib
[Ot2]{otal-2 J.-P. Otal. Sur la g\'eom\'etrie symplectique de l'espace des g\'eod\'esiques d'une vari\'et\'e \`a courbure n\'egative. Revista Matematica Iberoamericana 8 (1992), 441--456.
1503.01321.extracted_bib
[PT]{payne-thas S. E. Payne and J. A. Thas. Finite generalized quadrangles. European Mathematical Society (EMS), 2009. %
1503.01321.extracted_bib
[S]{S J. \'Swi\c{atkowski, Trivalent polygonal complexes of nonpositive curvature and Platonic symmetry. Geom Dedicata. 70 (1998), %pgs. 87--110. %
1503.01321.extracted_bib
[Th]{Th A. Thomas, Lattices in hyperbolic buildings. To appear in the volume ``Geometry, topology, and dynamics in negative curvature - Proceedings'', London Math. Soc., Lecture Notes series. Preprint available on the %arXiv:1204.0287
1503.01321.extracted_bib
[S]{sun Z. Sun. Marked length spectra and areas of non-positively curved cone metrics. Geom. Dedicata 178 (2015), 189--194.
1503.01321.extracted_bib
[vM]{van-Maldeghem H. van Maldeghem. Generalized polygons. Birkh\"auser Verlag, 1998.
1503.01321.extracted_bib
[We]{Weiss-structure R. M. Weiss. The structure of spherical buildings. Princeton University Press, 2003.
1503.01321.extracted_bib
[Wi]{W D.T. Wise, A flat plane that is not the limit of periodic flat planes. Algebr. Geom. Topol. 3 (2003), pgs. 147--154.
1503.01321.extracted_bib
[X]{X X. Xie, Quasi-isometric rigidity of {Fuchsian buildings. Topology 45 (2006), pgs. 101--169.