On Stability of Nodal $L_k^p$-Maps I
Gang Liu
1509.07187.tex
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\begin{document}

\title{On  Stability of  Nodal $L_k^p$-Maps I}
\author{\\
\\
\\
Gang Liu }
\date{May,  2015}
\maketitle



\section{Introduction}
In this paper and its sequels, we  introduce the  concept   of  weak stability for  general nodal $L_k^p$-maps as  a natural generalization of  stability for $J$-holomorphic maps;
 then give a complete characterization of  the weakly stable  nodal $L_k^p$-maps in term of  their isotropy   groups.
  Among  weakly stable $L_k^p$-maps, the stable ones are those whose isotropies are finite.  As a consequence, we prove   the  Hausdorffness,  without any stability
  assumption,  of the space of unparametrized
  nodal $L_k^p$-maps modeled on a fixed tree.
   %As a special case,   we  obtain  a reasonable definition of the "moduli space" of genus zero  nodal curves ( without marked points).  
   
   We will only deal with the genus zero case. This is justified since for a  nodal $L_k^p$-map, the  part of its  reparametrization group with positive dimension consists of the reparametrizations of  its  unstable genus zero components. In this paper, we only consider the  nodal $L_k^p$-maps modeled on a fixed tree $T$. The general cases
allowing the changes of the topological types of the domains   and   targets will be treated in  the sequels of this paper. We now describe the main results of this paper.

Fix a (minimal) label  $L$ of a given tree $T$ such that the labeled
tree ${\hat T}=(T, L)$ is stable. Let ${\cal M}_{\hat T}$ be the moduli space of  genus zero stable  curves with $n$ marked points modeled on ${\hat  T}$ and ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$ be the
universal curve over
${\cal M}_{\hat T}$. Here $n$ is the minimal number of marked points added to a genus zero nodal surface modeled on $T$ to make it stable. Fix a
Riemannian manifold $M.$
Let ${\tilde {\cal B}}^{\hat T}={\tilde {\cal B}}^{\hat T}_{k, p}$  be the set of $L_k^p$-maps $f:\Sigma\rightarrow M$ with $\Sigma$ being one of the fibers of the universal  family  ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$. One can show that ${\tilde {\cal B}}^{\hat T}_{k, p}$ is a Banach  manifold of class $[m_0]$ (see Section 2).  Here $m_0=k-2/p$ is the Sobolev differentiability of $f$.  Through this paper, we will alway assume that
 $[m_0]\geq 1$  so that each component of $f$ is at least of class $C^1.$ Roughly speaking, the space $ {\tilde {\cal  B}}^{\hat T}$ can be thought as the space of parametrized nodal $L_k^p$-maps modeled on $T$. Let $  {\cal  B}^{\hat T}$ be the space of equivalence classes of nodal $L_k^p$-maps. It can be obtained from $ {\tilde {\cal  B}}^{\hat T}$ as orbit space under the actions of the reparametrization groups $G_f$  of the elements $f$ in $ {\tilde {\cal  B}}^{\hat T}$. The normal subgroup  of $G_f$   preserving the components of $f$  is independent  of $f$ in the sense that they can be identified each other
 canonically. Denote the resulting group by $G_T$. Note that the quotient group $G_{f/T}=G_f/G_T$ is  a finite group that exchanges  the components of $f$. Thus $  {\cal  B}^{\hat T}$ can be obtained by first forming the global quotient $  {\tilde {\cal  B}}^{\hat T}/G_{T}$, then   quotient out a further locally finite equivalence relation by the actions of $G_{f/T}.$
 Note that the action of $G_T$ is continuous (see Sec. 2).

 \begin{theorem}
 	The space  $   {\tilde {\cal  B}}^{\hat T}$ is $G_T$-Hausdorff in the sense that for any two diffent $G_T$-orbits $G_Tf_1$ and $G_Tf_2$, there exit $G_T$-neighborhoods $G_TU_1$ and $G_TU_2$ such that $G_TU_1 \cap G_TU_2=\varphi$. Therefore, the global quotient $  {\tilde {\cal  B}}^{\hat T}/G_{T}$ is Hausdroff.
 \end{theorem}

 The proof of the above theorem implies  the following

 \begin{pro}
The space of unparametrized nodal $L_k^p$-maps, $  {\cal  B}^{\hat T}$  is always  Hausdroff.

\end{pro}


 Note that in general a nodal map  $f$ may have components, such as trivial unstable component, so that the istropy group ${\Gamma_f}$ is
 not compact. In this case, the action of $G_T$ on $   {\tilde {\cal  B}}^{\hat T}$ is certainly not proper so that $f$ is not stable in any reasonable sense. Yet, the above results show that when the topological type of the domains  is fixed given by $T$, the Hausdorffness of $  {\tilde {\cal  B}}^{\hat T}/G_{T}$  and  $  {\cal  B}^{\hat T}$ still holds  without  requiring any stability conditions. This seems to contradict to our experience in Gromov-Witten theory. Indeed, when
 the topological type of the domains  is allowed to change, the corresponding space ${\cal B} $ of unparemetrized nodal $L_k^p$-maps is not Hausdroff anymore. But this non-Hausdorffness occurs in a rather definite manner mainly caused by the appearance of the extra trivial bubbles obtained by a non-convergence sequence of reparametrizations. Once such  degenerations are prohibited, the-Hausdorffness will be restored even allowing the change of the topological types of the domains. In other words, for any subspace of  ${\cal B} $ which do not contain a sequence convergent to a nodal map with extra trivial bubbles, the Hausdorffness still holds.  The detail of this will be given in a sequel of this paper.

 Next we define the weak stability.

 \begin{definition}
  A nodal $L_k^p$-map is said to be weakly stable if non of its unstable component is a trivial map.	
 \end{definition}



 Let  ${\tilde {\cal  B}}^{ws}_{\hat T}$ be the subspace  of $ {\tilde {\cal  B}}^{\hat T} $  consisting of weakly stable  nodal $L_k^p$-maps. As before,   ${\tilde {\cal  B}}^{ws}_{\hat T}$ can be thought as the space of parametrized  weakly stable nodal maps modeled on $T$. Let  ${\cal  B}^{ws}_{\hat T}$ be the corresponding space of
 unparametrized  weakly stable nodal maps.

\medskip 
\noindent
 {\bf  Note:}   Applying  the above definition to   $J$-holomorphic  nodal maps, we get
   one of the standard definitions for  stable $J$-holomorphic maps
 in Gromov-Witten theory. %Put it differently, we get the weakly stability here if we replace the $J$-holomorphic maps in  (one of  ) the standard definition(s) of stable 
  %$J$-holomorphic maps by $L_k^p$-maps.
  In this sense the weak stability for $L_k^p$-maps here is a natural generalization 
of the stability for  $J$-holomorphic maps.  However, as we will see below, in general
the isotropy group $\Gamma_f$ for a weakly stable $L_k^p$-map $f$ is not finite but only compact. 

 \begin{theorem}
 	The action of $G_T$ on  $   {\tilde {\cal  B}}^{ws}_{\hat T}$ is
 	proper in the sense that  for any $f_1$ and $f_2$ in  $   {\tilde {\cal  B}}^{ws}_{\hat T}$,  there exist the corresponding  open neighborhoods $U_1$ and $U_2$ containing $f_1$ and $f_2$ respectively  and compact subsets $K_1$ and $K_2$ in $G_T$  such that  for any $h_1$ in $U_1$ ($h_2$ in $U_2$) and $g_1$ in $G_T\setminus K_1$  ($g_2$ in $G_T\setminus K_2$ ), $g_1\cdot h_1$ is not in $U_2$ ($g_2\cdot h_2$ is not in $U_1$).
 	
 	

 \end{theorem}

{\bf  Note:}  It was explained in [L1] that  for  the finite dimensional case,  the definition  of  properness  above is equivalent to the usual definition.

 \begin{cor}

 For any weakly stable nodal map $f$,  the  isotropy $\Gamma_f$ of the $G_T$-action or $G_f$-action  is always compact.
 \end{cor}

 In fact the same is true for the isotropy groups of the non-trivial components of a nodal $L_k^p$-map.






 {\bf  Note:}
 It follows from the continuity of the $G_T$-action that   $\Gamma_f$  is closed in  $G_T$ so that it is a  compact  Lie subgroup of $G_T$.






 \begin{cor}
 	
 	A  nodal $L_k^p$ map $f$ is weakly stable if and only if the  isotropy $\Gamma_f$ is compact.

 
 
 \end{cor}

\medskip 
This gives a complete characterization of weak stability of  nodal $L_k^p$-maps in term of their  isotropy groups.  
 




\medskip
 \begin{definition}
 	A weakly stable nodal $L_k^p$-map $f$ is said to be  stable if
 	it has no infinitesimal automorphisms.  In other words, the dimension
 	of the Lie algebra  $Lie(\Gamma_f)$ is equal to zero.
 \end{definition}


 Since $\Gamma_f$ is compact for a  weakly stable nodal map, the above condition is equivalent to the following definition.
 
 
 

\begin{definition}
	A weakly stable nodal $L_k^p$-map $f$ is said to be  stable if
its isotropy group 	$\Gamma_f$ is a finite group.
\end{definition}

Using the above characterization  for weakly stable nodal $L_k^p$-map, we get
 the following corollary.


\begin{cor}
	
	A   nodal  $L_k^p$-map $f$ is stable if and only if the  isotropy $\Gamma_f$ is finite.
	
\end{cor}



Recall that the corresponding characterization  for stable $J$-holomorphic maps is the following well-known proposition in GW theory.

\begin{pro}
	
	A $J$-holomorphic  nodal  map $f$ is stable if and only if the  isotropy $\Gamma_f$ is finite.
	
\end{pro}
 
 Thus stable $L_k^p$-maps and stable $J$-holomorphic    maps have exactly the same 
 characterizations in term of their isotropies. One the other hand, we have seen that
 in term of the definitions,  weakly stable $L_k^p$-maps are the natural generalizations of   stable $J$-holomorphic   maps. 
 
 \medskip
  \medskip
 Let
  $   {\tilde {\cal  B}}^{s}_{\hat T}$ be the  set  of stable nodal $L_k^p$-maps and $    {\cal  B}^{s}_{\hat T}$ be the corresponding
  quotient space of unparametrized stable nodal maps. Next theorem follows from the definition.

  \begin{theorem}
  	The actions of $G_T$ and $G_f$  on  $   {\tilde {\cal  B}}^{s}_{\hat T}$  are
  proper with finite istropies so that $   {\cal  B}^{s}_{\hat T}$ is
   a topological Banach orbifold.  Moreover, the action of $G_T$
   has local slices in $   {\tilde {\cal  B}}^{s}_{\hat T}$
    that provide local uniformizers for the orbiford structure
     of  $   {\cal  B}^{s}_{\hat T}$.   In particular, the subspace
      of  $   {\cal  B}^{s}_{\hat T}$ consisting of stable nodal
      maps with the  trivial isotropy is a topological Banach manifold.
  	
  \end{theorem}

  To get a better  understanding of the stability, we divide the unstable  components of  weakly stable maps into the following two classes.
  Let $f_v:\Sigma_v\simeq {\bf C P }^1\rightarrow M$ be such a component.
  Then $f_v$ is said to be $2$-dimensional if there is a point $x_0$ in
  $\Sigma_v$ such that the rank of $(df_v)_x$ is equal to two, otherwise it is one-dimensional.

  It follows from the definition that

  \begin{lemma}
  If all unstable components of a weakly stable map are
  	 $2$-dimensional,  it is stable.
  \end{lemma}

  Thus we  only need to consider  a nontrivial unstable component  $f_v:\Sigma_v\simeq {\bf C P }^1\rightarrow M$ of $f$ that is  one-dimensional. The reparametrization group of $f_v$ is the group
  $G_{\Sigma_v}$ considered as subgroup of  ${\bf PSL}(2, {\bf C})$ that fixes the double points of $\Sigma_v\simeq {\bf C P }^1$ (if there are any). Note that
  the group $G_T$ is the product of these reparametrization groups $G_{\Sigma_v}$ of the unstable
  components.
  Denote  the isotropy group inside $G_T$ by  $\Gamma_f^T$. Then  $\Gamma_f^T$
  is the product of the corresponding isotropy  groups $\Gamma_{f_v}^T\subset G_{\Sigma_v}$ of the unstable
  components.
  Then if the  dimension of  $\Gamma_f$  is  positive, so is $\Gamma_f^T$. This implies that the same is true for  some  $\Gamma_{f_v}^T$ with $f_v$ non-trivial.
  
 By  our assumption $\Gamma_f^T$ and  $\Gamma_{f_v}^T$ are compact.
  Since $f_v$ is nontrivial, the action of $\Gamma_{f_v}^T$ on $\Sigma_v$ can not be transitive (even for the case that $f:S^2\rightarrow M$) so that the compact subgroup $SU(2)$ of ${\bf PSL}(2, {\bf C})$  is not contained in $\Gamma_{f_v}^T$. Since $f_v$ is one-dimensional, the compactness of  $\Gamma_{f_v}^T$ then implies that  its identity component  $(\Gamma_{f_v}^T)^0$  is the standard  $S^1\simeq SO(2)\subset G_{\Sigma_v}$ upto conjugations. Let $\mu: {\bf C P }^1\rightarrow  {\bf so}(2)$  be the  moment map for the standard action of $S^1$ on ${\bf CP}^1.$ 
  Then we  have proved   the following lemma.

  \begin{lemma}
  	If $f_v: \Sigma_v\simeq {\bf C P }^1\rightarrow M$ is an one-dimensional unstable  component of a weakly stable map of class $C^1$, then upto a conjugation by the action of  $G_{\Sigma_v}$,
  	$f_v={\bar f}_v\circ \mu$. Here ${\bar f}_v:I=[-1, 1]\rightarrow M$ is a $C^1$-map,  and $\mu:\Sigma_v\simeq  {\bf C P }^1\rightarrow I\subset {\bf R}^1\simeq {\bf so}(2)$  is the "height" function above.
  	
  	 
  \end{lemma}


  Any one-dimensional map with the form in the above lemma (upto the action of reparametrizations) will be called a standard $S^1$-invariant map.

  \begin{lemma}
  	If non of the  unstable  component of a weakly stable map is standard $S^1$-invariant, then it is stable.
  	
  \end{lemma}

  A one-dimensional non-trivial  map $f_v: \Sigma_v\simeq {\bf C P }^1\rightarrow M$
   is said to be standard if $f_v={\bar f}_v\circ \pi$. Here ${\bar f}_v:I=[-1, 1]\rightarrow M$  and $\pi:\Sigma_v\simeq  {\bf C P }^1\rightarrow I\subset {\bf R}^1$ are two  $C^1$ maps.  Consier the special case that    $\pi:\Sigma_v\simeq  {\bf C P }^1\rightarrow  {\bf R}^1$ is a $C^1$ map with only two  critical points. Clearly any such $\pi$ is
    conjugate to the moment map $\mu$ by a differomorphism of $\Sigma_v.$


   \begin{lemma}
   	Even within the space of the standard one-dimensional $C^1$ maps with two critical points, the generic elements are not standard $S^1$-invariant maps.
   	
   \end{lemma}



  Thus we have large supplies of stable maps even all the unstable components are standard one-dimensional maps  with two critical points.


  The next proposition summaries the discussion so far on the continuous part of the isotropy
  group of  a nodal map.

  \begin{pro}
  The identity component  $\Gamma_f^0$ of the isotropy group of the nontrivial components of a	 nodal $L_k^p$-map  $f$ is a torus $T^n=(S^1)^n.$ In particular, $\Gamma_f^0\simeq T^n$ for a weakly stable $L_k^p$-map $f.$
  \end{pro}


\medskip
\noindent
{\bf Note}: In the finite dimensional case,  the slice theorem  for a Lie group $G$ acting  smoothly and properly on a  manifold $M$ 
states  that for any $m\in M,$ there is a diffeormorphism  from the disc  bundle
$G\times _{\Gamma_m}D$ onto a neighborhood of the orbit $G\cdot m$ in $M$. Here ${\Gamma_m}$ is the (compact) isotropy group of $m$.
The properness and the Sc smoothness of the $G_T$-action on ${\widetilde {\cal B}}^{ws}_{\hat T}$  implies  that the same  statement holds in the Sc setting.
The detailed proof  of this will be  given some where else.


\medskip
\medskip

 In the formulation of the nodal maps, we have suppressed the information about the homology classes $[f_v]\in H_2(M,{\bf Z})$ represented by of the components $f_v$.  Let $A_T=\{A_v\in H_2(M, {\bf Z}), v\in T\}$ be a collection of second homology classes associated to $T$. Denote the  space of nodal $L_k^p$-maps  modeled on ${\hat T}$ that represent class $A_T$  by ${\tilde {\cal B}} ^ {\hat T}_{A_T}.$

 \begin{pro} 	Assume that for all unstable component, the corresponding $A_v\not = 0.$ Then all elements of
 	${\tilde {\cal B}}^{\hat T}_{A_T}$ are stable.   Consequently the actions of $G_T$ and $G_f$ are proper with finite isotropies, and  the quotient space  ${ \cal B} ^ {\hat T}_{A_T}$ of the  unparametrized nodal maps of class $A_T$ is a topological Banach orbifold.





 \end{pro}



  This paper is organized as follows.

  Section 2 provides the  preliminaries used to define the space ${\tilde {\cal B}}_{\hat T}$ of $L_k^p$-maps modeled on a labeled tree ${\hat T}.$
  
  

  Section 3 proves the  main  theorems  here as well as  some related results.
  The results here generalize the corresponding ones in [L1] for $L_k^p$-maps with  domain $S^2$. 
  
  Section 4  gives   an elementary discussion on
  the  classification of the  group $S_T$ of the  automorphisms of a tree $T$.  A comparison of $S_T$  with the discrete part of the reparametrization  group of a nodal surface of type $T$ is made in this section, which gives   
   the constraints on the discrete part of the reparametrization  group.



   \section{Preliminaries}
   \subsection{Nodal curves, stable curves  modeled on ${\hat T}$}


   Recall that a tree $T$  is a connected 1-dimensional ( abstract)  simplicial complex without cycles. We still use $T$ to denote the set of its vertices. For the veritces $v$ and $u$ in $T$, the edge relation will be denoted by $uEv.$


   Given two trees $T_1$ and $T_2$, a map  $\phi:T_1\rightarrow T_2$  is said to a pre-morphism if the following  condition (1) holds: for any vertices $u$ and $v$ in $T_1$  with $uEv, $
   ether $\phi(u)=\phi(v)$ or $\phi(u)E\phi(v).$  A pre-morphism $\phi:T_1\rightarrow T_2$ is said to be a morphism if in addition the following condition 
 (2)  holds: for any $u_2$ in $T_2$ the inverse image $\phi^{-1}(u_2)$ is a subtree of $T_1$ (or empty).
   % An equivalent  condition of (2) is  (2') $\phi$ contracts subtrees.
  
  
   To see the meaning of the condition (2), consider the following
   simple example.
   %In other words, $\phi$ is a (generalized) simplicial map.
   % Next lemma shows that the weaker definition used here is equivalent to the usual definition in the literature of Gromov-Witten theory.

    {\bf Example:} Let $T_1=\{v_1, v_2, v_3\}$ with $v_1Ev_2, v_2Ev_3$ be the chain connecting $v_1$ and $v_3$, and $T_2=\{u_1, u_2\}$ with $u_1Eu_2$. Let $\phi:T_1\rightarrow T_2$ sending $\phi(v_1)=\phi(v_3)=u_1$ and $\phi(v_2)=u_2$.
    Then $\phi $ satisfies
    (1) (hence is a pre-morphism) but not (2).

     Note that  the two sides adjacent to  $v_2$ is flipped and identified by $\phi.$  In general,  a  pre-morphism   $\phi:T_1\rightarrow T_2$   is said to have
      flipped identifications if there is a subchain of length two in $T_1$ that is gotten flipped and  identified  into a single edge at its middle vertex under ${\phi }$.

  \begin{lemma}
   	 A pre-morphism  $\phi:T_1\rightarrow T_2$  is a morphism if and only if it does not contain any flipped identifications.
   \end{lemma}

   \proof

    Let $C_1$ and $C_2$ be two connected components of  ${\phi}^{-1}(u)$ in $T_1$.   Chose a shortest chain $C(v_1, v_2)$ from $v_1\in C_1$ and $v_2\in C_2$ connecting the two components.
   	Since $\phi (v_1)=\phi(v_2)=u,$  ether the  image $\phi(C(v_1, v_2))  $ is  a cycle that is impossible or the map $\phi: C(v_1, v_2)\rightarrow T_2$ contains a flipped identifications.
   	
   	
 %The lemma then follows from above  observation.

\QED

  The above condition (2) is justified in Gromov-Witten theory since for the gluing construction at a double,  the induced  map on corresponding trees
  does not contain flipped identifications so that  it is a morphism.

More specifically,  given a edge $[vu]$ at $v\in T$,  let $T_{v;u}$ be the tree obtained by contracting the edge into the vertex $v$. The corresponding map denoted by  $\phi_{v;u}: T\rightarrow T_{v;u}$
  will be called a  contracting  map (of $[vu] $ )at $v$.  It is a surjective morphism.

   Note that   such maps  generate
  all surjective morphism from a tree $T$ in the sense that any surjective morphism $\phi; T\rightarrow T_1$ can be factorized as
  $\phi=\rho \circ \phi'$. Here $\rho:T\rightarrow T'$ is composition of a sequence of above basic edge-contracting maps, and $\phi':T\rightarrow T$  is an isomorphism.


  The full meaning of the condition (2) will become clear in the lemma of this section on the compatibility of the total orders of the two $n$-labeled stable  trees
   with their underlying
    trees being related by a morphism defined above.



  As usual,  the notion of morphism  gives rise the notion of isomorphisms between trees. An automorphism of $T$ is a self isomorphism.

   A vertex $v$ is said to unstable if its valence $Val(v)\leq 2.$
    An unstable vertex $v$  is said to be a tip of the tree  $T$  if $Val(v)=1.$

   \begin{lemma}
   	 An isomorphism $\phi:T_1\rightarrow T_2$ is determined by its  action on the tips of $T_1$.
   \end{lemma}

   \proof

   Argue by  induction on the numbers of edges.
    The starting point is the tree with only one edge  and two vertices.

    Note that in this case, the condition (2) is automatically true.
    So we start with a one-to-one map $\phi: T_1\rightarrow T_2$ between the vertex sets that satisfies condition (1). Let  $v_1\in T_1$ be a tip of $ T_1$ and $u_1\in T_1$ is the vertex next to $v_1$. Denote their images in $T_2$ under $\phi$ by  $v_2$ and $u_2$.
    Let $T_1'=T_1\setminus [v_1u_1] $ and consider $\phi':T_1'\rightarrow T_2'=\phi(T_1').$ By induction $\phi'$ is determined by its values on the tips of $T'_1$.

     If $Val( u_1)>2,$
    tips of $T'_1$ are also the tips of $T_1$ so that $\phi$ is determined by its values on the tips.

    If $Val( u_1)=2$, the $u_1$ becomes a tip in  $T_1'$, but the value $\phi'(u_1)$ is already determined by $\phi: [v_1u_1]\rightarrow [v_2u_2]$. Hence in this case by induction $\phi'$ is determined by the values of $\phi$ on the tips of $T_1$.





    \QED




   \begin{lemma}
   	{
   	Let 	$\phi_{v;u}: T\rightarrow T_{v;u}$ be  the  contracting  map of edge  $[vu] $ at $v$. Then 	$\phi_{v;u}$ is determined by its values on the tips. }
   \end{lemma}

   \proof

   { Choosing a tip $v_1\not = v$ and edge $[v_1u_1]$ of $T$, and   consider $T'=T\setminus [v_1u_1] $ and  $T'_{v;u}=T_{v;u}\setminus [v_1u_1] $. Let  $\phi':T'\rightarrow T_{v;u}'.$ By induction $\phi'$ is determined by its values on the tips of $T'$.  The rest of the argument is the same as the one in the
   	previous lemma.}

    \QED

    \begin{cor}
   A morphism $\phi: T_1\rightarrow T_2$ is determined by its values on the tips of $T_1$.	
    \end{cor}

    {\bf Example:}  Consider again  the example with
    $T_1=\{v_1, v_2, v_3\}$ being   the chain  $[v_1, v_2, v_3]$  of length two   and $T_2=\{u_1, u_2\}$   with  the edge $[u_1u_2]$.  Let  $\phi:T_1\rightarrow T_2$  with  $\phi(v_1)=\phi(v_3)=u_1,$  $\phi(v_2)=u_2$ and $\psi (v_i)=u_1, i=1, 2,3$ be two maps. Then both maps have the same values on the tips of $T_1$ and satisfies condition (1)  in the definition of  morphism. This shows that  the condition (2) in the definition of  morphism is crucial for the above   corollary.





   An ordered tree $T_o$ is a pair $(T, O) $ consisting of a tree $T$ with an order $O$ for its tips.  The above lemma implies

   \begin{lemma}
   	Any order preserving automorphism of an ordered tree is the identity map.
   \end{lemma}


    \begin{lemma}
    	An order on the tips of $T$ gives rise a  total  order to all the vertices and edges described in the proof below. Let $\phi_{v;u}:T\rightarrow T_{v;u}$ be a contracting map at the edge $[vu]$.
    	In the case that $v$ is the initial tip of $T$,  assume  that it is still so in $T_{v;u}$.
    	Then the total order of $T$ obtained above naturally induces
    	an order of the tips of $T_{v;u}$  as well as the  total order
    	of $T_{v;u}$  such that $\phi_{v;u}:T\rightarrow T_{v;u}$ is "order preserving"
    	in the obvious sense.
    	Consequently, a surjective morphism $\phi:T\rightarrow T_1$ carries  the 
    	total order on $T$ defined above to the corresponding 	total order on $T_1$
    	such that the map $\phi$ is order preserving.
    	 
    \end{lemma}

    \proof

We definition of the total order induced by $T_o$  first.

 Let $\{t_0, t_2, \cdots, t_l\}$ be the ordered set of tips of $T$.
 Consider the unique directed chain $[t_0, t_1]$. This gives rise an order for all vertices and edges lying on the directed chain.
    Assume that the union $T^{k-1}$ of the first $k-1$ such  directed connecting chains has totally ordered. Note that intersection of two  connecting chain $[t_0t_m]$ and $[t_0t_k]$ is  a chain of the form $[t_0v]$ (including $t_0=:[t_0t_0]$). Hence for a fixed $k$ the intersection  $T^{k-1}\cap [t_0t_k]$ is a
   subchain of $[t_0t_k]$ of the form  $[t_0v_k]$. Hence part of the $k$-th  directed connecting chains,  $[t_0v_k]$ is already totally ordered from that of $T^{k-1}$.  Then  rest part of $[t_0t_k]$,  $(v_kt_k]$ (with $v_k$ not included ) is totally ordered in the obvious manner so that $T^{k}$ is  totally ordered.

   
   To prove the last part of the lemma, note that either the connecting chain $[t_0v]=[t_0u][uv]$ or $[t_0u]=[t_0v][vu]$.  We may assume that $[t_0v]=[t_0u][uv]$. Then there is a smallest $k\geq 1$ such that 
   $[t_0uv]$ is the initial part of the chain $[t_0t_k].$ Then $T^{k-1}=T^{k-1}_{v;u}$ so that they have the same total order. If $v\not= t_k$, 
   then $t_k\in T^{k}_{v;u}$ so that the set of tips of $T^{k}$ and $T^{k}_{v;u}$
   is the same. It is easy to see that in this case there is a compatible total order for  $T^{k}_{v;u}$ which extends to a total order on $T_{v;u}$ compatible with 
   $\phi_{v;u}$.
   
      The  remaining  case is that  $v=t_k$. In this case,   $t_k=v$ is not a tip of $T_{v; u}$ anymore. It is replaced by $t'_{k}=u$ with   the  connecting  chain $[t_0t'_k]$ in $T_{v; u}$.
      Then    total order of $T^{k}$  gives the  compatible  total order $T^{k}_{v;u}$ in this case.  As before, the  extension of the total order from $T^{k}$ to $T$ gives the induced compatible  extension  of  the total order from $T^{k}_{v;u}$ to $T_{v;u}$.
     
      
  

   Note that in above argument, one can choose any vertex rather than a tip  $t_0$ as the  initial one.
    \QED





    An $n$-labeled tree ${\hat T}=(T, L)$  consists of   a tree $T$ and a  label $L$ which is a map $L: {\bf n}=\{1, 2, \cdots, n\}\rightarrow
    T.$

A morphism $\phi:(T_1, L_1) \rightarrow (T_2, L_2)$ between
two $n$-labeled trees is defined to be a morphism $\phi:T_1\rightarrow T_2$  of the underlying trees such that $\phi\circ L_1=L_2.$

    An  $n$-labeled tree ${\hat T}=(T, L)$ is said to be stable if  for any  $v\in T$,  $Val(v)+\#(L^{-1}(v))\geq 3$.

    Given an   stable  $n$-labeled   tree ${\hat T}=(T, L)$, on each tip $t$ of $T$, let $i\in {\bf n}$ be the smallest   number in   $L^{-1}(t)\subset {\bf n}.$ We label $t$ by $t_i$ so that  all the tips of $T$ are ordered. Then above lemma shows that the tree $T$ is totally ordered. This total order can be extended into a total order of ${\hat T}$ including all the subsets $L^{-1}(v), v\in T$ as follows.
     %that give  the decomposition of $ {\bf n}$ for  the definition of $L$.
     %This  extended total  order is  defined    following.
      For each vertex $v$,  elements in  $L^{-1}(v)$   are  considered as  "edges" (loops) connecting $v$ to itself and ordered by the natural order on  $L^{-1}(v)$.

     % With this interpretation of the total order of  $n$-labeled stable  trees, the  statements similar  to the above lemma hold for stable  labeled trees.

   % \begin{lemma}
    %	Given a contracting  morphism $\phi_{v; u}:{ T}\rightarrow { T}_1$ with 
    	%$T$ being  the underlying tree of an $n$-labeled stable tree ${\hat T},$
    %	then  there is an unique $n$-labeled stable tree
    %	${\hat T}_1$ on  ${ T}_1$  such that $\phi_{v; u}$ is a morphism 
    	%between the  $n$-labeled stable trees ${\hat T}$  and  ${\hat T}_1$.
    	 % Moreover it is compatible with  the two total orders on $T$ and $T_1$ induced from the corresponding labels.  	
    	%Consequently, a surjective morphism $\phi:T\rightarrow T_1$ carries any stable $n$-label $L$  on  $T$  to the corresponding label on $T_1$ denote by $L_1$  such that the corresponding labels induces
    %	compatible total orders on $T$ and $T_1$ with respect to $\phi$.
   % \end{lemma}

Now consider a  morphism $\phi:(T_1, L_1) \rightarrow (T_2, L_2)$ between
two  stable $n$-labeled trees  with the underlying  morphism $\phi:T_1\rightarrow T_2$  being surjective. Then the label $L_1$ determines an total order on $T_1$, and hence a compatible total order on $T_2$ as well by above lemma. However this induced total order on $T_2$  map not be the same  as  the one induced from $L_2$ even $L_2=\phi\circ L_1.$   As an extreme case of this, assume that $T_2$ is a single point. In general  induced order on the set ${\bf n}$ from the total order on $T_1$ determined by $L_1$ may not be the natural order for ${\bf n}$. On the other hand, the induced order  on  ${\bf n}$ by $L_2$  is just the   natural order.

 




    A morphism $\phi:(T_1, L_1) \rightarrow (T_2, L_2)$ between
    two $n$-labeled trees is said to be an equivalence  if it is an isomorphism of the underlying trees. An self equivalence of ${\hat T}$ is called an automorphism.

     If $\phi$  an automorphism of a stable $n$-labeled tree ${\hat T}=(T, L),$ then for any tip $t$,  $i_t\in L^{-1}(t)\not =\varphi$,  and  $t=L(i_t)=\phi\circ L (i_t)=\phi (t)$ so that
     $\phi $ is  identity. In other words, there is no nontrivial
     automorphism for a stable $n$-labeled tree.


On the other hand,  there is a weaker notion  of morphism between $n$-labeled trees.  An unordered    morphism $\phi:(T_1, L_1) \rightarrow (T_2, L_2)$ between
   two $n$-labeled trees is a morphism $\phi:T_1\rightarrow T_2$ and a permutation $p\in S_n: {\bf n}_1\rightarrow {\bf n}_2$ such that $\phi\circ L_1=L_2\circ p.$  The unordered  equivalence and automorphism can be defined in the same fashion as before.

   It is easy to see that the group of unordered automorphisms  of ${\hat T}$ is the same as the group of automorphisms  of the underlying tree  ${ T}.$ In other words, for unordered automorphisms  of labeled trees, the labeling by $L$ does not impose any further constraints.






Given a tree $T$, a minimal stabilization of $T$ is an  $n$-labeled stable  tree ${\hat T}=(T, L)$  such that on each unstable vertex $v\in T$, $Val(v)+\#(L^{-1}(v)= 3.$ There are finitely many such minimal
 stabilizations corresponding to the possible "stable maps" $L$ from ${\bf n}$ to $T$. Note that for a minimal $n$-stabilization, $n-3=\#(E )+\Sigma_{v\in T, \, stable}(\#({\bf d}_v)
 -3).$
 Here $\#(E)$ is the number of edges of $T$ which is half of the number of elements in the edge relation $E$ of $T$.

    A genus zero nodal curve modeled on $T$ is the pair ${\bf \Sigma}=(\Sigma, {\bf d})$. Here to  each $v\in T$ we associate  a component $\Sigma_v$
     of $\Sigma$, that is   a genus zero Riemann surface, and ${\bf d}=\cup_{v\in T} {\bf d}_v$ where  ${\bf d}_v=\{d_{uv}, uEv\}$ is the set of double points on $\Sigma_v$.
    Then $\Sigma$ is the underlying nodal surface obtained by identifying the corresponding double points,
    $\Sigma=\coprod_{v\in T}\Sigma_v/\{{d_{uv}=d_{vu}, uEv}\}.$


  ${\bullet}$  {\bf Note on notations:} The double points  ${\bf d}$ and  ${\bf \Sigma}$ (or $\Sigma=:\Sigma_{\bf d}$)
   determine each other. Later on if there is no confusion, we will use  any of them to denote the nodal surface. Similar remark is applicable to stable curves defined below.

    A genus zero  (stable)  curve modeled on an $n$-labeled (stable) tree  ${\hat T}$ is the tuple  ${\bf \Sigma}=(\Sigma, {\bf d}, {\bf x})$.
    The double points set ${\bf d}$ is defined same as above for a nodal
     curve. The set of marked points, $ {\bf x}=\cup_{v\in T}{\bf x}_v$, where ${\bf x}_v=\{x_{vi}, L(i)=v\}$ is the set of marked points on $\Sigma_v$. Let ${\bf p}_v= {\bf d}_v\cup  {\bf x}_v $ be set of special points on $\Sigma_v$. The stability condition implies that on each component  $\Sigma_v$, $\#({\bf p}_v)\geq 3.$

     Like the case of the nodal curves, the underlying
     stable curve, denoted by $\Sigma$ is obtained by identifying the corresponding double points.


    %The order of the marked points of an $n$-labeled stable tree ${\hat T}$ gives an order of the tips of $T$ so that all the vertices and edges of $T$  together with the marked points are ordered altogether. Note that in this new  order, the marked points are reordered in such a way that first according to the order of the vertices that  marked points lie on then within a vertex, the marked points are ordered by there "natural" order.

   % Now the gluing operation at a singe vertex $v\in T$ gives rise a morphism $\phi_v$ from a $n$-labeled stable tree ${\hat T}$ to another $n$-labeled stable tree ${\hat T}_1.$  Next lemma will not be used in the main body of this paper. We only state it without
    % proof.






  A map $\phi: \Sigma_1\rightarrow \Sigma_2$ is said to be equivalence between two nodal surfaces  ${\bf \Sigma}_1=(\Sigma_1, {\bf d}_2)$ and ${\bf \Sigma}_2=(\Sigma_2, {\bf d}_2)$ modeled on $T_1$  and $T_2$ respectively  if it is a homeomorphism such that (1)$\phi_v:(\Sigma_1)_v\rightarrow (\Sigma_2)_{\phi (v)}, v\in T_1$ is biholomorphic. It follows that the condition (2) holds:  $uE_1v$  if and only if  $\phi (u)E_2 \phi( v)$  so that the induced map $\phi:T_1\rightarrow T_2$ is an equivalence, and  $\phi((d_1)_{uv})=(d_2)_{uv}$ for $uE_1v$ and $\phi (u)E_2 \phi( v).$
  	
  	Such an equivalence can also be defined as a collection of maps $\phi_v, v\in T$ that satisfies condition (1) and (2) above.
  	
  	 Thus by  renaming the vertices of $T_2$, we only need to consider
  	  equivalence of two nodal surfaces  modeled on the same tree $T$.

  This leads to the following more restrictive  definition for marked nodal surfaces or stable curves.

   Given two $n$-marked nodal curves  (or stable curves) ${\bf \Sigma}_1$  and   ${\bf \Sigma}_1$ modeled on $n$-labeled trees ${\hat T}_1$ and ${\hat T}_2$
    respectively with the same underlying tree $T$, an equivalence $\phi: \Sigma_1\rightarrow \Sigma_2$ between the underlying nodal surface  is said to be equivalence between the  two $n$-marked nodal curves if
    $\phi ({\bf x}_1)={\bf x}_2$ such that the induced map $\phi:{\hat T}_1\rightarrow {\hat T}_2$ is an equivalence of the $n$  label trees.

  The case  relevant to   next subsection is that ${\hat T}_1={\hat T}_2={\hat T}.$ In this case,  an equivalence between  two marked nodal surfaces modeled on ${\hat T}$  induces an automorphism $\phi: {\hat T}\rightarrow {\hat T}.$ We will show  later in this section  that for stable curves,  any such induced  automorphism  is trivial.






  \subsection{Moduli space of the  genus zero stable curves modeled on ${\hat T}$}
The moduli space of the  genus zero stable curves modeled on ${\hat T}$, denoted by ${\cal M}_{\hat T }$, is defined to be the equivalence classes of such curves.

 Note that
  given a genus zero  stable curve
  modeled on an $n$-labeled tree ${\hat T},$  the induced total order by ${\hat T}$ on the special points ${\bf p}_v$ of $\Sigma$ gives rise a
  well defined biholomorphic identification $\phi_v:\Sigma_v\rightarrow
  {\bf CP}^1, v\in T$ by sending the first three special points on $\Sigma_v$ to $0, 1$ and $\infty$.   Thus ${\cal M}_{\hat T }$ is the quotient space of ${\tilde {\cal M} }_{\hat T }$, as an open set of the product of  ${\bf CP}^1$ with $|{\bf p}|$ factors,  by the obvious diagonal action of $H_T=\prod_{v\in T} {\bf PSL}(2,{\bf C })_v$.
  Here $|{\bf p}|$ is the number of spacial points, and ${\tilde {\cal M} }_{\hat T }$ is the set of special points ${\bf p}=\{p_{vi}, v\in T\}$
  with $p_{vi}$ being special points on $ {\bf CP}^1_v(=$ a copy of $ {\bf CP}^1$).

  It is easy to see that the action of $H_T$ is free and holomorphic.  One can show that the action is proper so that  ${\cal M}_{\hat T }$
  is a complex  manifold. An other way to show that  ${\cal M}_{\hat T }$ is a complex manifold is to use the slice of the $H_T$-action
  given by a complex submanifold of ${\tilde {\cal M}}_{\hat T }$ described in next lemma.

  In order to describe the universal curve as well as the gluing construction used in the sequels of this paper, we introduce the "global coordinate"  of  ${\cal M}_{\hat T }$ defined by multi-cross ratios selected by the order of $p_{vi}$ given by ${\hat T}.$


 As mentioned above, each point in ${\tilde {\cal M}}_{\hat T }$ can be consider as a tuple of the special points on the components ${\bf CP}^1_v, v\in T.$  For each $v\in T,$ the special points $p_{vi}$ are ordered by ${\hat T}$ described before given by the index $i$. For simplicity, we may assume that $i=1, 2, \cdots, I_v.$ Then the "coordinate" for $p_{vi}$ with $3<i\leq I_v,$ is given by the  cross-ratio $w_{vi}=:w_{v123i}=(p_{v1}:p_{v2}:p_{v3}:p_{vi}).$ The tuple of all such coordinates together,  denoted by $w_{\bf p},$  with the order given by $i$ is the coordinate
  for the tuple ${\bf p}=:\{p_{vi}, v\in T\}$ of special points.  The proof of the next lemma  is clear.

  \begin{lemma}
  	The map ${\bf p}\rightarrow w_{\bf p}$is $H_T$-invariant. It gives rise a
  	global  "coordinate  chart"  for  ${\cal M}_T$. The complex submanifold of ${\tilde {\cal M}}_{\hat T}$, $S{\tilde {\cal M}}_{\hat T }=\{p_{vi}, v\in T\, |p_{v1}=0, p_{v2}=1, p_{v3}=\infty\}$ is a slice of the $H_T$-action. The global coordinate map above is naturally defined on the slice.
  \end{lemma}

 Given above slice $S{\tilde {\cal M}}_{\hat T }=\prod_vS_v,$ we  define (a model of) the universal curve as a family ${\cal U}_{\hat T }\rightarrow S{\tilde {\cal M}}_{\hat T }$ as follows. For each $v\in T $ we  define ${\cal U}_{v  }\rightarrow S_{v }$  to be the trivial family  ${\cal U}_{v  }= S_{v }\times {\bf CP}^1_v \rightarrow S_v$ first. Note  that for fixed $i$, the tautological map $p_{vi}\in S_v\rightarrow
 p_{vi}\in S_v$ is a holomorphic section of the family so that it gives
 rise a divisor in ${\cal U}_{v  }$.   Now  for each fixed $v\in T, $ the pull-back of ${\cal U}_{v  }\rightarrow S_{v }$  by the projection map $S{\tilde {\cal M}}_{\hat T }\rightarrow S_v$ gives a family over $S{\tilde {\cal M}}_{\hat T }$, still denoted by 
 ${\cal U}_{v  }\rightarrow S{\tilde {\cal M}}_{\hat T }$.
 Then the normalization of the universal family  ${\cal U}_{\hat T }\rightarrow S{\tilde {\cal M}}_{\hat T }$ to be defined  is the disjoint union of these  ${\cal U}_{v  }\rightarrow S{\tilde {\cal M}}_{\hat T }, v\in T$. 
 
 Now  the desired universal curve   ${\cal U}_{\hat T }$ is obtained from ${\cal U}_{v  }$ by identify those divisors that are corresponding to the tautological sections  from double points.  This implies that  ${\cal U}_{\hat T }$ is an analytic space with only normal crossing singularities.



  \subsection{The space of  nodal maps modeled on $T$}
  Fix a Riemannian manifold $(M, g_M) $.
  Let $\Sigma$ be a nodal surface modeled on $T$.
   A  continuous map $f:\Sigma\rightarrow M$ map  is said to be a nodal $L_k^p$-map modeled on $T$
  if each its component $f_v:\Sigma_v\rightarrow M, v\in T$ is of class
  $L_k^p$. Here the $L_k^p$-norm is measured with respect the metric $g_M$ on $M$  and  the pull-back of the Fubini-Study  metric on $\Sigma_v$  by a  identification $\phi_v:\Sigma_v\simeq {\bf CP}^1.$  Note that $\phi_v$ depends on the choice of a minimal stabilization of $\Sigma $ but the induced $L_k^p$-norms are equivalent with respect to different choices.

  Two such nodal maps $f_1:\Sigma_1\rightarrow M$ and $f_2:\Sigma_2\rightarrow M$ are said to be equivalent if  is an equivalence map $\phi: \Sigma_1\rightarrow  \Sigma_2$ such that $f_1=f_2\circ \phi.$   Thus using the identifications of the components of the domains given by $\phi_v:\Sigma_v\simeq {\bf CP}^1$, we may assume that each component of a nodal map has the form $f_v:\Sigma_v={\bf CP}^1_v\rightarrow M.$ Again, here $\phi_v$ depends on a particular choice of a minimal stabilization of $\Sigma$ which can be obtained by fixing a minimal stable labeling ${\hat T}$ of $T$.
  This leads to the following definition.

  \begin{definition}
  	Fix a $n$-labeled tree ${\hat T}$ that is a minimal stabilization of the given tree $T$. Let ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$ be the universal family of genus zero stable curves with $n$-marked points. The space of genus zero stable $L_k^p$-maps modeled on ${\hat T}$ is defined to be
  	$${\tilde {\bf B}}^{\hat T} =\{({\bf \Sigma}, f)|\, f=\{f_v:\Sigma_v\rightarrow M, v\in T\},\, f_v(d_{vu})=f_u(d_{uv })\, \, if\, \, uEv, \, \|f_v\|_{k, p}<\infty\}.$$
  	Here the domain $\Sigma$ of $f$ with components $\Sigma_v, v\in T$ is the underlying nodal curve of the stable curve ${\bf \Sigma},$ which  is  a fiber of the universal family ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$.
  \end{definition}

   The space ${\tilde {\bf B}}^{\hat T}$ can be thought as a   precise version of the intuitive notion of the space of  parametrized
    nodal maps.

  \begin{pro}
  ${\tilde {\bf B}}^{\hat T}$ is a Banach manifold of class $[m_0].$
  \end{pro}	

  \proof

  Recall that the universal curve can be realized as ${\cal U}_{\hat T}\rightarrow S{\cal M }_{\hat T}.$ The slice $S{\cal M }_{\hat T}$ is the product $\prod_{v\in T}S_v,$ where $S_v=\{p_{vi}\}$ is the set of special points on ${\bf CP}^1_v$ with  the first three of them are $0, 1, \infty.$ Note that for an unstable $v$, $S_v$  has only one element $(0, 1, \infty).$
  	
  	Let $${\tilde {\cal B}}_v=\{(p_v, f_v))|\, p_v\in S_v, \, f_v:{\bf CP}^1_v\rightarrow M , \, \|f_v\|_{k, p}< \infty\}.$$ Then ${\tilde {\cal B}}_v$ is naturally identified with $S_v\times {\tilde {\cal B}}({\bf CP}^1_v)$. Here ${\tilde {\cal B}}({\bf CP}^1_v)$ is the  Banach manifold of $L_k^p$ maps with the  fixed domain ${\bf CP}^1_v$. This implies that  ${\tilde {\cal B}}_v$ and hence
  	$\prod_{v\in T}{\tilde {\cal B}}_v$ is the Banach manifold.
  	
  	\begin{pro}
  The evaluation map $E_v:{\tilde {\cal B}}_v\rightarrow M^{|{\bf d_v}|}$given by $(p_v, f_v)\rightarrow f_v(d_v)$ is of class $[m_0].$ Here $d_v$ are the double points among the special points $p_v$ and $f_v(d_v)$  is the tuple of the evaluations of $f_v$ at the  doubles  of  $d_v$  considered   as an element in  $M^{|{\bf d_v}|}$. Moreover, the evaluation map is surjective submersion.
  	\end{pro}
  This proposition is proved in [L] and [L?]. 

  Let $E=\prod_{v\in T}E_v:{\tilde {\cal B}}_v\rightarrow M^{|{\bf d}|}$
  be the total evaluation map at double points. Then  it is still a submersion so that it is  transversal to the diagonal $\Delta_T\subset  M^{|{\bf d}|}.$ Here $\Delta_T=\{ m_{uv}=m_{vu}\in M, when \, \,uEv \, \,for \, \, u, v \in T  \}$ is a submanifold of $  M^{|{\bf d}|}=\{m_{uv}\in M, uEv\}.$  This implies that ${\tilde {\bf B}}^{\hat T}=E^{-1}(\Delta_T)$ is a Banach manifold of class $[m_0].$

  \QED


  In Gromov-Witten theory we are mainly interested
  in the space of equivalent classes of the nodal $L_k^p$-maps, denoted
  by ${\cal B}^T.$  It can be obtained by quotient out the action of the reparametrization groups on
  ${\tilde {\bf B}}^{\hat T}$.   These actions are induced from the corresponding  actions on the domains. We now spill out more details on these actions.

Let ${\Sigma}  =\Sigma_f$ be the domain  of a nodal map modeled on the tree $T$ with a fixed minimal stabilization  modeled on a $n$ labeled tree ${\hat T}$.  Recall that the order on the vertices and edges of $T$ from ${\hat T}$ gives rise the identifications of each component $\Sigma_v, v\in T$ with  ${\bf CP}^1_v.$   For an unstable component  ${\bf CP}^1_v$ with one or two double points ${\bf d}_v$, we will assume that ${\bf d}_v$ are in the standard position $0$ or $0$, $\infty.$
Then the reparametrization group $G_v=G_{\Sigma_v}$ is defined to be the
subgroup $G_i, i=1,2$ of $G_0={\bf PSL}(2, {\bf C})$ consisting of the biholomorphic maps of ${\bf CP}^1_v$ that fix the double point(s) the standard ${\bf d}_v.$ Let $G_T$ be the product $\prod_v G_v$ with $v$ being unstable components.
Then (I) Since  ${\bf d}_v$  are in their standard positions in the unstable components, there is no "moduli" coming from such double pints. This  implies that $G_T$ is independent of $({\Sigma}, {\bf d}, {\bf x})\in {\cal M}_T$, that justifies our notation.  (II) Any element in $G_v$ is determine by its action on the corresponding (minimal )marked point(s) ${\bf x}_v.$

 Note that any element $\phi\in G_T$ can be considered
 as an automorphism of $\Sigma$ by requiring that $\phi_v=$ identity on any stable component $\Sigma_v$. Let $G_f=:G_{\Sigma}$ be the group of reparametrization of the nodal curve $\Sigma$ consisting of all self identifications.  Clearly
 $G_{\Sigma}$ depends on $\Sigma.$  This dependence comes from the following finite subgroups.

 Let  $(\Sigma, [{\bf  x}])$ be marked
 nodal surface with unordered marking $[{\bf x}]$
 obtained from ${\bf x}.$ A  map $\phi:(\Sigma_1, [{\bf  x}]_1)\rightarrow :(\Sigma_2, [{\bf  x}]_2)$  is said to be an equivalence between the two such surfaces if $\phi$ is an equivalence
 of the underlying nodal surfaces such that $\phi$ identifies  ${\bf x}_1$ to
 ${\bf x}_2$   sets (without any order).

 Denote the finite group of  automorphisms (=self equivalence) of $(\Sigma, [{\bf  x}])$ by  $G_{\Sigma, [{\bf  x}]}$. Note that here
 $(\Sigma, {\bf x})$ is a minimal stabilization of $\Sigma$ so that each
 top bubble (=unstable component $\Sigma_v$ with $v$ being a tip of $T$) has two marked points with order in ${\bf x}.$ Let
 $G^0_{\Sigma, [{\bf  x}]}$ be the subgroup of  $G^0_{\Sigma, [{\bf  x}]}$ consisting automorphisms that preserve the order of the two marked points of each top bubble. For each tip $v\in T$, let  ${\bf Z}_{2, v}\simeq  {\bf Z}_{2}$ and ${\bf Z}_{2, T}\prod_{v=\, tip }   {\bf Z}_{2, v}$ considered as subgroup of $G_{\Sigma, [{\bf  x}]}$.
 Here ${\bf Z}_{2, v}$  is the group of involution of $(\Sigma_v, [{\bf  x}]_v)$ that exchanges
 the two marked points in ${\bf x}_v$ for a tip $v\in T.$
 Note that $G^0_{\Sigma, [{\bf  x}]}$  is a normal  subgroup of  $G_{\Sigma, [{\bf  x}]}$, and $G_{\Sigma, [{\bf  x}]}$ is the semi direct product of $G^0_{\Sigma, [{\bf  x}]}$ and ${\bf Z}_{2, T}.$

 %Here (A) The finite group is the  group of the and (B)
 % can be described in   the following lemma whose proof is clear.

 \begin{lemma}
 	The group $G_T$ is the normal subgroup of $G_{\Sigma}$ consisting of  self identifications that preserve the components of $\Sigma.$
 The quotient group $G_{f/T}=:G_f/G_T$ is a finite group that switches the components of $\Sigma.$  The exact sequence $1\rightarrow G_T\rightarrow 	G_{\Sigma}=G_{f}\rightarrow 	G_{f/T}\rightarrow 1$  splits in the sense that (I) $G_T$ and $G^0_{\Sigma, [{\bf  x}]}$ generate 	$G_{f/T}$ and (II) The induced homomorphism $G^0_{\Sigma, [{\bf  x}]}\rightarrow G_{f/T}$ is an isomorphism.
 %The subgroup  $G^0_{\Sigma, [{\bf  x}]}$ is also a normal.
 Therefore  $G_{\Sigma}$ is  the semi direct product of $G_T$ and $G^0_{\Sigma, [{\bf  x}]}$.

 %surjective with the kernel being the product
 %of ${\bf Z}_{2, v}$ for all $v$ being tips of $T$.

 \end{lemma}

 \proof

 %Only the last statement require a proof.
 First note that (i) the $G^0_{\Sigma, [{\bf  x}]}\cap G_T=\varphi;$
 (ii) $G_T$ and  $G^0_{\Sigma, [{\bf  x}]}$ generate  $G_{\Sigma}$.

 Next  proposition shows that any element $\phi$ in $G_{\Sigma}$ is determined by its effect on the marked points ${\bf x}$. Given $\psi\in G^0_{\Sigma, [{\bf  x}]}$, assume that for a top bubble $\psi_v: (\Sigma_v;x_{v1}, x_{v2})\rightarrow (\Sigma_{v'};x_{v'1}, x_{v'2}).$ Let $\gamma=\{\gamma_v, v\in Tip(T)\}\in G_T$. It is easy to check that   $\gamma_{v'}\circ \psi_v=\psi_v \circ\gamma_v$ for $v, v'\in Tip(T)$ above,  and  that the same relation holds for the  unstable components with only one marked point. Since $G_T$ acts trivially on the stable components,  this implies that  $G_T$ is normal.

  
 \QED



To prove next proposition,  we need a lemma on  existence of special level one points on a tree. The tips of a tree $T$ will be called  level zero points. Given a  vertex
 $v\in T$ that is not a tip, it  is said to be a level one point if there is a tip $u\in T$ such that (i) there is a simple chain $[uv]$ connection $u$ and $v$; (ii)  $[uv]$ is a maximal simple chain.  A  chain connecting $u$ and $v$ is said to be simple if all the intermediate vertices have valence two. Note that the condition (ii) above implies that the valence of $v$ is at least three.
 Clearly a tree $T$ does not have level one points if and only if $T$ is a simple
  chain.

 \begin{lemma}
 	Assume that $T$  has level one points. Then 	there exits  a level one point $v\in T$  such that there exist two tips $u_1$ and   $u_2$,  and the corresponding  maximal simple chains $[u_1v]$ and $[u_2v]$ connecting $u_1$,  $u_2$  to $v$.
 \end{lemma}

 \proof

 Give  the tips of  $T$ an order so that $T$ is totally ordered.
Then the first vertex $v$ on the last chain (connecting to the last tip) is the sought-after level one point. 

 
 
 \QED

  \begin{pro}  Any   element $\phi\in G_{\Sigma}$is determined by its effect on the unstable components.
  \end{pro}
  \proof

Note that in the  case that the tree $T$ has no level points,  $T$ itself  is a simple chain.   Then  any component of a nodal curve modeled on $T$  is unstable so that the lemma is automatically true in this case.

Therefore we only need to consider the case that $\Sigma$ has level one points. Let $\phi$ be an automorphism of such a nodal surface  $\Sigma$.

Let $\Sigma_v$ be a component   with $v\in T$ be the terminal vertex of a maximal simple chain $[uv]$ for a tip $u\in T.$ Thus $v$ is a level one vertex. We may assume that $v$ satisfies the conclusion of the above lemma so that there is another maximal simple chain $[u_1v]$ with tip $u_1\in T$. Then by the assumption $\phi$ is already defined on all the corresponding components $\Sigma_w$ for any vertices $w\not = v$ lying on any of these two simple chains. Now   remove all components
other than $\Sigma_v$ along the chain $[uv]$. If $\Sigma_v$ is stable in the new nodal surface $\Sigma'$ then all the unstable components in  $\Sigma'$ remain unstable in $\Sigma$ so that $\phi$ is defined on these components. By induction, $\phi$ is defined on $\Sigma'$.This implies the lemma.

Thus we may assume that $\Sigma_v$ is not stable in $\Sigma'$. Since $Val(v)\geq 3$ in $T$, this can only happen if  $\Sigma_v$ has one and  only one more double point $d_v$ other than the two double points
coming from the simple chains $[uv]$ and $[u_1v]$. Note  $\phi$  maps
the  double points to the double points on the corresponding components. By assumption,  the values of $\phi$ at two of the double points on $\Sigma_v$  that  joins to the two components on the two maximal simple chains are already determined,  lying on  the corresponding
component $\Sigma_{v'}$ as two of its three double points. Then  $\phi$  maps the third  double point of $\Sigma_v$ to the third one of $\Sigma_v'$ so that the map $\phi$ is determined on $\Sigma_v$. Now we are in the position  to apply  the  induction for this case too.

 
  \QED


 \begin{cor}
 The automorphism group $G_{\Sigma, {\bf x}}$ of a stable curve $(\Sigma, {\bf x}	)$
 	 is trivial.
 	
 \end{cor}

 The group $G_{\Sigma, [{\bf  x}]}$ can be regarded as the isotropy group of the marked nodal surface $(\Sigma, [{\bf  x}])\in {\cal N}_{n,T},$ where ${\cal N}_{n,T}$ is the moduli space of genus zero nodal curves with (unordered ) $n$-marked points modeled on ${\hat T}.$


 %Next lemma shows that any element $\phi$ in $G_{\Sigma}$ is determined by its effect on the marked points ${\bf x}$.
% Let  $G^0_{\Sigma, [{\bf  x}]}$ be the normal  subgroup $G_{\Sigma, [{\bf  x}]}$ of index two,
  %whose elements  preserve the order of the two marked points on each of top bubble components.  Then $G_{f/T}\simeq  G^0_{\Sigma, [{\bf  x}]}$, and above exact sequence splits so that  $G{\Sigma}=:G_f$ is the semi direct product of $G_T$ and $G^0_{\Sigma, [{\bf  x}]}$.

\begin{pro}
	
	The action of the  isotropy group $G_{\Sigma, [{\bf  x}]}$ on the "central fiber" $ (\Sigma, {\bf  x})$ can be extended into an action on  the  underlying   curves  of the universal family
	${\cal U}_{{\hat T}}\rightarrow S{ \cal M}_{\hat T}$ preserving the unordered marked point sets. The extended action above is compatible
	with the action of $G_T$ so that these actions together gives rise a
	fiber preserving  action of  $G_{\Sigma}$ on the universal curve ${\cal U}_{{\hat T}}\rightarrow S{ \cal M}{\hat T}$ that is an
	extension of the action on the fiber $ (\Sigma, {\bf  x})$.
	
	Moreover, on the base  $S{ \cal M}{\hat T}$, the action of $G_T$ is trivial while the action of  $G_{\Sigma, [{\bf  x}]}$ on an sufficiently small open neighborhood $\Lambda(\Sigma, {\bf  x})$ of  $(\Sigma, {\bf  x})$ in  ${\cal M}_{T}\simeq
	S{ \cal M}{\hat T}$ gives rise a local  uniformizer of ${\cal N}_{n,T}$. Here ${\cal N}_{n,T}$ is the moduli space of the stable nodal curves with n unordered marked points.
In particular, for any $(\Sigma', {\bf  x'})\in \Lambda(\Sigma, {\bf  x})$, the action of $G_{\Sigma, [{\bf  x}]}$ induces a surjective
group homomorphism $G_{\Sigma, [{\bf  x}]}\rightarrow G_{\Sigma', [{\bf  x'}]}$  which is compatible with their actions on the base.  Consequently, on the neighborhood  $\Lambda(\Sigma, {\bf  x})$,   any  $ G_{\Sigma', [{\bf  x'}]}$-orbit  of a given point  is covered by its $ G_{\Sigma, [{\bf  x}]}$-orbit.


	
%The action of the  isotropy group $G_{\Sigma, [{\bf  x}]}$  can be extended into an action on a local  uniformizer of ${\cal N}_{n,T}$ near  $(\Sigma, [{\bf  x}])$. In fact  the local  uniformizer here    is an open neighborhood of  $(\Sigma, {\bf  x})$ in  ${\cal M}_{T}$;    the action   of the  isotropy group $G_{\Sigma, [{\bf  x}]}$ on the neighborhood gives rise the local orbifold structure of   ${\cal N}_{n,T}$. Moreover, there is a further compatible "local"  extension of the above action to   the group $G_{\Sigma}$  acting on above open neighborhood of  $(\Sigma, {\bf  x})$ in  ${\cal M}_{T}$. This local extension can be prolonged into an action of $G_{\Sigma}$. Further more, these actions of $G_{\Sigma}$ from a functorial system	in the same way as the system  of the actions of $(\Sigma, [{\bf  x}])$ on open  neighborhoods of  $(\Sigma, {\bf  x})$ in  ${\cal M}_{T}$ used to define
%the orbifold structure for   ${\cal N}_{n,T}$.

\end{pro}

We prove the following corollary first.

\begin{cor}
	The local action of  $G_{\Sigma}$ on the  open neighborhood $\Lambda(\Sigma, {\bf  x})$ of  $(\Sigma, {\bf  x})$ in  $
	S{ \cal M}_{\hat T}$ defined  above is a local model for the "moduli
	space" of nodal curves of type $T$ in the sense that  for    any two stable curves  as two fibers over 	$\Lambda(\Sigma, {\bf  x})$, their underlying nodal curves  are equivalent  if and only if they are in the same  $G_{\Sigma}$-orbit.
\end{cor}

\proof

For any two stable curves $({\Sigma'}_1, {\bf x'}_1)$ and $({\Sigma'}_2, {\bf x'}_2)$ in the local universal family over $\Lambda(\Sigma, {\bf  x})$, their underlying  nodal curves are equivalent if and only if one of them,  $({\Sigma'}_2, {\bf x'}_2)$	for instance, is in the $G_{{\Sigma'}_1}$-orbit.
Since   $G_{{\Sigma'}_1}$ is generated by $G_T$ and  $G_{{\Sigma'}_1, {[\bf x'}_1]} $, up to a $G_T$-action, $({\Sigma'}_2, {\bf x'}_2)$	 is in the $G_{{\Sigma'}_1, {[\bf x'}_1]} $-orbit.


Since $G_T$ acts trivially on the base  $\Lambda(\Sigma, {\bf  x})$ and  its actions on base  are identical when  it  is regarded  either as a subgroup of   $G_{{\Sigma'}_1}$ or as a subgroup of  $G_{{\Sigma}}$. Hence we may assume that the induced identification of the two corresponding points on  the base is given by an element in $G_{{\Sigma'}_1, {[\bf x'}_1]}.$


Now  by last part of the above proposition, on the base the  $G_{\Sigma'_1,  {[\bf x'}_1]}$-orbit of
$({\Sigma'}_1, {\bf x'}_1)$ is covered by its $G_{\Sigma,  {[\bf x}]}$-orbit.
% implies that for any $(\Sigma', {\bf  x'}_1)\in \Lambda(\Sigma, {\bf  x})$, the induced homomorphism of the group of  reparametrizations,  $G_{\Sigma,  {[\bf x}] }\rightarrow G_{\Sigma'_1,  {[\bf x'}_1]}$  is surjective  so that .
%	Here  	$({\Sigma'}, {\bf x'})$ is only considered as a point in base.  
However, it follow from the definition that the actions of these two finite symmetry groups come from the corresponding  ones acting on  the universal family preserving the set of unordered marked sections.  Hence
as curves (fibers) in the  universal family $G_{\Sigma'_1,  {[\bf x'}_1]}$-orbit of
$({\Sigma'}_1, {\bf x'}_1)$ is still covered by its $G_{\Sigma,  {[\bf x}]}$-orbit. This proves that the original identification is realized, upto
a $G_T$ action,   by an action of an element in $G_{\Sigma,  {[\bf x}]}$ so that
$({\Sigma'}_1, {\bf x'}_1)$ and $({\Sigma'}_2, {\bf x'}_2)$   are in the same  $G_{\Sigma}$-orbit.

%A functorial version of the universal property of the universal family implies  that the two actions can be lifted into   the local universal family so that $({\Sigma'}, {\bf x'})$-orbit of $({\Sigma'}, {\bf x'})$ is still covered by its $G_{\Sigma}$-orbit.


%Now for any two stable curves $({\Sigma'}_1, {\bf x'}_1)$ and $({\Sigma'}_2, {\bf x'}_2)$ in the local universal family over $\Lambda(\Sigma, {\bf  x})$, their underlying  nodal curves are equivalent if and only if one of them,  $({\Sigma'}_2, {\bf x'}_2)$	for instance, is in the $G_{{\Sigma'}_1}$-orbit.




\QED




 The key ingredient to prove above proposition is the universal
 property of the universal curve.

   For simplicity we assume that $(\Sigma, {\bf x})$ is still minimally stable. This is sufficient for the purpose of this paper.

\noindent ${\bullet}$ Universality of ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$.

The above propositions follows from  the universality of ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}$ in the proper category of family of stable curves. In algebraic geometry, the  usual formulation of a good family  uses  the algebraic notion of flat family of such curves.
%It is well-know that in higher genus case, due to the nontrivial isotropies,  the universal  does
In the genus zero case, the tautological family of stable curves with $n$ marked points, $n\geq 3$, ${\cal U}_{0, n}\rightarrow {\overline {\cal M}}_{0, n}$ is  indeed an  universal flat family.  Since flatness is preserved under the base changes, any other flat family of such curves can be obtained by pull-backs.

In this paper and its sequels, we are working on   the  categories
of smooth or complex manifolds. The existence of the universal family above suggests that    good families  of genus zero n-stable curves  in  these categories can be simply defined to be the pull-backs of the universal family.


%For  the case at our hands that $T$ is fixed, this can be described
%more specifically as follows.

%Recall  that ${\cal U}_{\hat T}$  can be recovered from  its the normalization $\coprod_{v\in T} {\hat {\cal U}}_v $ by gluing along the divisors ${\hat D}_{v}$ of the set of double points. Here  $\coprod_{v\in T} {\hat {\cal U}}_v $ is the pull-back  of ${\cal U}_v\simeq S_v\times {\bf CP}^1_v\rightarrow S_v$ by the projection $ \prod_vS_v\rightarrow S_v.$
%Since ${\hat T}$ is a minimal stabilization, on each unstable $v$, $S_v$ is just  the "single point" $(0, 1, \infty)$ under $S_v\simeq {\bf CP}^1_v$,  and for a stable $v$, $S_v =D_v$ (with first three fixed as the standard ones). Hence ${\cal M}_{\hat T}$ as the slice $S{\tilde {\cal M}}_{\hat T}=:\prod_vS_v\simeq \prod_vD_v$. The image of  the  diagonal sections $D_v\rightarrow
%D_v\times D_v\subset {\cal U}_v=S_v\times {\bf CP}^1_v$ together gives rise the divisor of the set of double points,  denoted by  ${\hat D}_v$ above.

\begin{definition}

A smooth
(holomorphic) family of nodal curves on a smooth (complex) manifold $M$  modeled on $T$ or ${\hat T}$   is defined to be (obtained by )
a smooth (holomorphic) map $h:M\rightarrow S{\tilde {\cal M}}_{\hat T}=:\{h_v:M\rightarrow S_v, v\in T \}.$ Here ${\hat T}$ is one of the (finitely many) minimal stabilizations of $T$.
\end{definition}


    Thus  the    smooth  or  holomorphic  family of nodal curves ${\cal C}=:{\cal C}_h\rightarrow M$ (and the corresponding family of  n-stable curve$({\cal C}_h, {\bf x})\rightarrow M$),  modeled on ${\hat T }$, are obtained
by pull back by $h$  of the  underline  universal family of  curves (together  with the $n$-marked sections). 

Now we  specify the meaning of the universality  in this setting so that it is not just a  tautology.

 To this end, let $(\Sigma, {\bf x})$ be "central fiber" ${\cal U}_{s_0}$ of  the  family  ${\cal U}_{\hat T}\rightarrow {\cal M}_{\hat T}=:S{\tilde {\cal M}}$. Here $s_0=\{s_{0, v}\in S_v, v\in T\}$ is the set of  special points on the central fiber.
Fix $m_0\in M$  with $h(m_0)=s_0$.  The central fiber ${\cal C}(m_0)$ of the family ${\cal C}\rightarrow M$   defined by  $h$  is just a copy of the central fiber $(\Sigma, {\bf x})$ of the universal family.
Let ${ \phi}:{\cal C}(m_0)\rightarrow {\cal U}_{s_0}$ be the identification of the two fibers.  Note that
${\hat \phi}$ as identification  of 
$n$-labeled stable curves  is unique.

Then the universal property  in this case is the claim that above identification of the central fibers  has an unique extension $\Phi:({\cal C}, {\cal C}(m_0))\rightarrow ({\cal U}_{\hat T}, {\cal U}_{s_0})$  that preserves the marked sections.


Of course with the respect  to  the  family  $({\cal C}, {\cal C}(m_0))\rightarrow (M, m_0)$ of the type given by a fixed   label tree ${\hat T},$ the universality for $({\cal U}_{\hat T}, {\cal U}_{s_0})\rightarrow (S{\tilde {\cal M}}, s_0)$ stated this way  is a tautology except the uniqueness of $\Phi$.




 

However, if we relabel the marked points by a different minimal stable label tree ${\hat T}_1$ and consider the same underlying family $({\cal C}, {\cal C}(m_0))\rightarrow (M, m_0)$ but with  the order of the marked points given by ${\hat T}_1$,  then  the universality with respect all possible  minimal stable labeling for  the underlying tree ${T}$ is not an tautology anymore.  What needs to be proved is stated 
in the following proposition.

\begin{pro}
Given a family  $({\cal C}, {\cal C}(m_0))\rightarrow (M, m_0)$ of type ${\hat T}_1$ and the family $({\cal U}_{\hat T}, {\cal U}_{s_0})\rightarrow (S{\tilde {\cal M}}, s_0)$ of type ${\hat T} $ with an equivalence $\phi: {\cal C}(m_0)\rightarrow {\cal U}_{s_0}$ as stable curves with ordered marked points, then there exists an unique extension of $\phi$, $\Phi:({\cal C}, {\cal C}(m_0))\rightarrow ({\cal U}_{\hat T}, {\cal U}_{s_0})$  that preserves the marked sections.


\end{pro}


Since the families ${\cal C}\rightarrow M$ is defined by pull-backs from the universal ones, the above proposition is equivalent to the following.

\begin{pro}
	Let  $({\cal U}_{{\hat T}_1}, {\cal U}_{s_1})\rightarrow (S{\tilde {\cal M}}, s_1)$  be the family obtained from the universal family
	$({\cal U}_{\hat T}, {\cal U}_{s_0})\rightarrow (S{\tilde {\cal M}}, s_0)$ of type ${\hat T} $ by relabel its double point sections given by ${\hat T}_1$. Assume that there is a self
	   equivalence $\phi:  {\cal U}_{s_1} \rightarrow {\cal U}_{s_0}$ as stable curves with ordered marked points with respect to ${\hat T}_1$ and ${\hat T}$ respectively, then there exists an unique extension of $\phi$, $\Phi:({\cal U}_{{\hat T}_1}, {\cal U}_{s_1})\rightarrow ({\cal U}_{\hat T}, {\cal U}_{s_0})$  that preserves the marked sections.
	
	
\end{pro}



 %gives the double point set ${\bf d}_v(m)$ on the component ${\cal C}_{v}(m)$ of the curve
%${\cal C}(m)$. Here ${\cal C}(m)$ is the fiber of ${\cal C}$ at $m$ that is obtained by gluing together ${\cal C}_{v}(m)\simeq {\bf CP}^1_v$ and ${\cal C}_{u}(m)\simeq {\bf CP}^1_v$ at the pair of  double points $h_{vu}(m)=h_{uv}(m),vEu$. Note that




%are determine each other

The proof the  above  proposition
is essentially same as the special case that  $s_0=s_1$.
We only give  the proof for this  case,  which can be
 restated as the following lemma.



\begin{lemma}
The action of the  isotropy group $G_{\Sigma, [{\bf  x}]}$ on the "central fiber" $ (\Sigma, [{\bf  x}])$ can be extended into an action on  the  underlying   curves of the  universal family
  ${\cal U}_{{\hat T}}\rightarrow S{ \cal M}_{\hat T}$ preserving the unordered marked point sets.
\end{lemma}




%one are using
%Perhaps a more  conceptual way to  see this is the following.
%The local model of the universal family ${\cal U}(\Sigma, {\bf x})\rightarrow {\Lambda}(\Sigma, {\bf x})$ is still a local universal family
%if we given the marked point set ${\bf x}$ a different order such as the one given by ${\bf x'}$ with ${ x'}_i=\phi( x_i).$  Then
%$\phi$ becomes a identification of the central fibers of the two family.
%
%The universality of the two family implies that there is a fiberwise extension of $\phi $ between the two  families,  $\Phi: {\cal U}(\Sigma, {\bf x})\rightarrow {\cal U}(\Sigma, {\bf x'})$. The restriction of $\Phi$ to each fiber,  $\phi_{\beta}:{\phi_{\beta}}:(\Sigma_{\beta}, {\bf x})\rightarrow (\Sigma_{\beta'}, {\bf x'})$ is the desired extension.





 \proof

  First  extend the isotropy action of   $G^0_{\Sigma, [{\bf  x}]}$ on the central fiber  to  the local universal family  ${\cal U}(\Sigma, {\bf x})\rightarrow \Lambda (\Sigma, {\bf x})$ as a fiberwise
 analytic action. We will use $\beta\in \Lambda (\Sigma, {\bf x})$ to denote the local parameter of the base of the universal family, which describes
  the locations (upto the actions of ${\bf PSL}(2, {\bf C})$), the "moduli"  of double points on stable components with respect to the reference double points on $\Sigma$.  Note that when ${\hat T}$ is fixed, using the total ordering  induced by ${\hat T}$,  there is a  coordinate describing the  global  "moduli"  of double points, 
  given by the multi cross-ratios with respect the first three
  double points on each stable component.
  The corresponding fiber will be denoted by $ (\Sigma_{\beta}, {\bf x}_{\beta})$ with the central fiber $(\Sigma, {\bf x})=(\Sigma_{{\beta}_0}, {\bf x}_{\beta_0})$. Note that here we have used the fact that $(\Sigma, {\bf x})$ is minimally stable.


  The desired action of $G^0_{\Sigma, [{\bf  x}]}$ on ${\cal U}(\Sigma, {\bf x})\rightarrow \Lambda (\Sigma, {\bf x})$ is defined as follows.

  For $\phi\in  G^0_{\Sigma, [{\bf  x}]}$ assume that the induced map   on $T$ given by   $v\rightarrow v'$ for any $v\in T$ so that $\phi_v:
  (\Sigma_v, {\bf p}_v)\rightarrow (\Sigma_{v'}, {\bf p}_{v'}).$

  Denote the  corresponding map to be defined on the fiber   ${\Sigma_{\beta}}$ by $\phi_{\beta}$. First note that for any $v\in T$ and $\beta$ near $\beta_0$, the component $\Sigma_{\beta, v}\simeq \Sigma_{\beta_0, v}=\Sigma_v$ with canonical identifications given by ${\hat T}.$  On each component, $\phi_{\beta, v}$ is defined to be $\phi_v$ with domain $\Sigma_{\beta, v}= \Sigma_{ v}$ and target  $\Sigma_{\beta', v'}= \Sigma_{ v'}$ for some $\beta'$ to be defined.
 Using these maps $\phi_{\beta, v}$, target  ${\Sigma_{\beta'}}$ is  obtained by gluing the components $\Sigma_{v'}$ along the corresponding double points in the obvious manner. More specifically, for each  double $d^{\beta}_{uv}=d^{\beta}_{vu}$  on  $\Sigma_{\beta}$ joining $\Sigma_{\beta, u}$ and $ \Sigma_{\beta, v}$,
  let $d^{\beta'}_{u'v'}$ and $d^{\beta'}_{v'u'}$ be the images $\phi_{\beta,u}(d^{\beta}_{uv})$ (=$\phi_{u}(d^{\beta}_{uv})$ by the definition above ) and $\phi_{\beta,u}(d^{\beta}_{vu})$,  considered as a double point on $\Sigma_{u'}$ and  $\Sigma_{v'}$ respectively. Then the surface $\Sigma_{\beta'}$ is defined to be the collection of the  components $\Sigma_{u'}$ joining together at  double points $d^{\beta'}_{u'v'}=d^{\beta'}_{v'u'}$ according to the condition $u'Ev'$ ($\Leftrightarrow uEv$). Clearly, by the construction, these maps $\phi_{\beta,v}, v\in T$ together define the desired  $\phi_{\beta}:(\Sigma_{\beta}, [{\bf x}])\rightarrow (\Sigma_{\beta'}, [{\bf x}]).$
  The parameter $\beta'$ here is defined as follows.


Since $\phi$ preserves the set of double points of $\Sigma, $    $d_{u'v'}=\phi_u(d_{uv}), uEv$ as sets.
If $\beta$ is close to $\beta_0$, the double points  $d^{\beta'}_{u'v'}= \phi_u(d^{\beta}_{uv}), uEv$ of the surface  $\Sigma_{\beta'}$  are  contained  the poly-discs  centered at double points
$d_{u'v'}=\phi_u(d_{uv}), uEv$ of $\Sigma$.   Thus with respect to these "centers" $(d_{uv}, uEv)$, the locations of the double points  $d^{\beta'}_{u'v'}$ (upto the obvious ${\bf PSL}(2, {\bf C})$ action on each component) are described by the  unique  parameter $\beta'$.  This defines the  $\beta'$.
Again, here we have used the fact that $(\Sigma_{\beta'}, {\bf x})$ has no extra marked points so that  $\beta'$ serves as a local coordinate of the moduli space ${\cal M}_T.$

Now both $\beta$ and $\beta'$ can be considered as the multi-cross ratios
of the double points $d^{\beta}_{uv}$  and $d^{\beta'}_{u'v'}$ with respect to  first three of them on each of the stable components using the same ordering  induced from  the same ${\hat T}$.  On the other hand, by the definition of $\phi_{\beta}$, the parameter $\beta$ is the same as the multi-cross ratios
of the  $d^{\beta'}_{u'v'}$ with respect to  first three of them on each of the stable components but using the  ordering of $T$  induced from  the new label $\phi\circ L$.  Thus both  $\beta$ and $\beta'$ are (part of all ) multi-ratios
of the very same  double point sets  $d^{\beta'}_{u'v'}$ selected with respect to two different orderings. Sine both of them already describe the full local moduli,  the implicit function theorem implies that $\beta'$ is a holomorphic function of $\beta, $ and vice versa for $\beta$ sufficiently close to $\beta_0$.

As mentioned before, when ${\hat T}$ and $\phi$  are fixed,  as the multi-cross ratios, $\beta$ and $\beta'$ are defined globally on $S{\cal M}_{{\cal T}}.$ Hence 
 $\beta'$ is still a holomorphic function of $\beta $  over $S{\cal M}_{{\cal T}}.$
 This defines the desired global 
  extended action $\Phi=\{\phi_{\beta}\}$ on the base $S{\cal M}_{{\cal T}}.$ Now using this together with the above existence and uniqueness  of local extensions, starting from the central fiber, we get the  desired global 
  extended action $\Phi=\{\phi_{\beta}\}$ on the whole  universal family  ${\cal U}_{{\hat T}}\rightarrow S{ \cal M}_{\hat T}$ by the usual argument of "analytic continuation."
 


%\in \Lambda (\Sigma, {\bf x})$ as

This finishes  the proof for $G^0_{\Sigma, [{\bf x}]}$. The proof  for  $G_{\Sigma, [{\bf x}]}$  is obtained  as  a special case of the argument below.
\QED

It is this   special form of universality that  is the key step  of  the proof below for  the main proposition before  on extension of the $G_{\Sigma}$-action.

${\bullet}$ Proof of the proposition  on extension of the $G_{\Sigma}$-action:

The action of $G_T$ is defined on any $\Sigma_\beta$ automatically. Indeed, the action of $G_T$ on   any stable component  is defined to be identity, and since  any unstable component $\Sigma_{\beta, v}$ is the  same as $\Sigma_{\beta', v}$, the action of $G_T$ on each such  $\Sigma_{ v}$ automatically gives the corresponding action on $\Sigma_{\beta, v}$.

Thus we already have  the extended actions of the two subgroups
$G_T$ and $G^0_{\Sigma, [{\bf  x}]}$. Recall that $G_{\Sigma}$ is the semi direct product of  $G_T$ and  $G^0_{\Sigma, [{\bf  x}]}$. More specifically, it is generated by   $G_T$ and  $G^0_{\Sigma, [{\bf  x}]}$ with the only relation   $\gamma_{v'}\circ \psi_v=\psi_v \circ\gamma_v, v\in T$ for $\gamma_v, \gamma_{v'}\in G_T$ acting on
$\Sigma_v$ and $\Sigma_{v'}$,  and $\psi_v:\Sigma_v\rightarrow \Sigma_{v'}$
 in $G^0_{\Sigma, [{\bf  x}]}.$
 	
 By the definition of the extended actions, on  a stable component
 of $\Sigma_\beta$, the above relation is automatically true for the extended actions since 	$\gamma_v$ and $ \gamma_{v'}$ are the identity map. One can verify that it is still true on any unstable component.




 \QED







%Recall that
%is the "coordinate" of the   moduli of the double points  The double points  $\phi_u(d^{\beta}_{uv}), uEv$ on the surface  which were already denoted as $d^{\beta'}_{u'v'}.$

%Note that the double points $\phi_u(d^{\beta}_{uv}), uEv$ are  holomorphic   in $\beta$. So  is ${\beta'}$.

  %Then  on each component   $v\in T$


   %unstable, denote  $\phi({\bf x}_v)$ by ${\bf x }_{v'}$ so that $\phi_v:(\Sigma_v, {\bf x}_v)\rightarrow (\Sigma_{v'}, {\bf x}_{v'})$
 % preserving the orders of marked points. Now  for  $\beta \in \Lambda (\Sigma, {\bf x})$, under the "canonical" identification between $\Sigma$ and  $\Sigma_{\beta}$
   %for all unstable $v\in T$,   $({\Sigma_{\beta}}_v, {{\bf x}_{\beta}}_v)$ is identical with  the component $(\Sigma_v, {\bf x}_v)$ in the central fiber.
  %Hence for each unstable $v\in T$, $\phi_v$ becomes a map denoted by ${\phi_{\beta}}_v$ with the domain $({\Sigma_{\beta}}_v, {{\bf x}_{\beta}}_v)$ and the target that is supposed to be $ ({\Sigma_{\beta'}}_{v'},{ {\bf x}_{\beta'}}_{v'})\simeq (\Sigma_{v'}, {\bf x}_{v'})$ for $\beta'=\beta'(\beta)$  as a holomorphic function of   $\beta$. The claim is that there is such a function $\beta$ such that  ${\phi_{\beta}}_v$   defined for unstable  $v$  so far  can be extended an identification ${\phi_{\beta}}:\Sigma_{\beta}\rightarrow \Sigma_{\beta'}$ as an element of  $G^0_{\Sigma, [{\bf  x}]}.$
 %The   claim  can be prove by induction in  the same way  to the proof
 %of the above proposition using the special level one point there.

 %Indeed, if $v\in T$ is a special level one point connecting by two maximal  simple chains $[uv]$ and $[u_1v]$, then like the above lemma, there are two cases to be considered.

 %The  first case is that $\Sigma_v$ has exact three double points. Then $\Sigma_v$  can be consider as  the corresponding  component of $\Sigma_{\beta})$. Let $\Sigma'_{\beta}$ be the stable curve obtained by remove all the components other than $\Sigma_{v}$ on the chain $[uv]$. Here the double point of $\Sigma_v$ from the removed chain  is regarded as a new marked point in $\Sigma'_{\beta}$.  Then the moduli of the double points in  $\Sigma_{\beta})$ and  $\Sigma'_{\beta})$ are given by the same multi cross-ratios denoted by $\beta$ that justifies our notation. Now $\Sigma_v$ becomes a unstable, but $\phi$ extends over it automatically.
% Then  by induction $\phi$  can be extended as required over $\Sigma'_{\beta}$, which is compatible with the definition of $\phi=\phi_{\beta}$ on the components on the chain $[uv].$


  %The   second  case is that $\Sigma_v$  is still stable after removing all the components other than $\Sigma_{v}$ on the chain $[uv]$ together with the double point on $\Sigma_{v}$ from the chain.
  %Let $\Sigma'_{\beta'}$ be the new stable curve. By induction $\phi$ extends into ${\phi_{\beta'}}$ with domain  $\Sigma'_{\beta'}$. We need to show that ${\phi_{\beta'}}$ and $\phi$ over the components of the chain $[uv]$  can be made compatible into the desired $\phi_{\beta}$.


  %Then $\phi_{\beta}$ is defined on
 %We leave the detail to the  reader.



  % On unstable components, $\phi_{\beta}$ is defined to be the same as $\phi$  by the "canonical" identifications of these components. By the above proposition $\phi$ is already determined by its action on these unstable components. This means that the values of $\phi$  at all double points While the induced action of $\phi$ of the base $\Lambda(\Sigma, {\bf x})$



 % Part of the lemma  concerning the actions  of $G_{\Sigma, [{\bf  x}]}$ of follows from the definition of ${\cal N}_{n,T}$ as orbofold.

 % It follows from next lemma that  for any nodal surface $\Sigma_{\beta}$ with minimal stabilization $(\Sigma_{\beta},
 % {\bf x}_{\beta})$ near the given $(\Sigma, {\bf x})=(\Sigma_{{\beta}_0}, {\bf x}_{{\beta}_0})$,any automorphism


 % the fixed a for any element

  %To  extend the action from $G_T$ and $G_{\Sigma, [{\bf  x}]}$ to  $G_{\Sigma}$ locally, we  note that for $\phi$ in $G_T$ or  $G_{\Sigma, [{\bf  x}]}$, it is determined by its effect on the first
  %three special points $\phi(p_{v1}), \phi(p_{v2}), \phi(p_{v3}), v\in T$.
 % Note that for a fixed $v\in T,$ the images of these three  points are lying  on the same component. In the case that $\phi\in G_T$, for a unstable component, we may assume that the first one or two points are double points. Then these double points are fixed by $\phi$, and  $\phi_v$ is determined by its effect on marked points $x_{vi}$ with any prescribed values $y_{vi}$.

  % Given  an element $\phi\in G_{\Sigma, [{\bf  x}]}$, let $\phi (x_{vi})=x{v'i}$ for an unstable component $v\in T$.


   	% The existence of such $\phi$ that send

%Note that the local uniformizer  here is an open neighborhood of  $(\Sigma, {\bf  x})$ in  ${\cal M}_{T}$
%Thus for a fixed $\Sigma$, the action generated by $G_T$ and $G_{\Sigma, [{\bf  x}]}$ covers the action $G_{\Sigma'}$ for nearby ${\Sigma'}$ as elements in    ${\cal M}_{T}$.

 We will assume the next proposition whose proof will be given in a forth coming paper.
 
\begin{pro}
The action map $G_T\times {\bf B}^{\hat T}\rightarrow 	 {\bf B}^{\hat T}$ is continuous. Moreover, for a fixed $g$ in $ G_T$ or  $G_{\Sigma}$,  the map ${\Phi}_g:{\bf B}^{\hat T}\rightarrow 	 {\bf B}^{\hat T}$ is a smooth map  of class $[m_0].$
Similar results hold for the actions of  $(\Sigma, {\bf  x})$.
\end{pro} 	


In next section, we will consider the induced action of  $G_f=:G_{\Sigma_f}$ on  the $G_f$-orbit of a prescribed neighborhoods $U_f\subset {\tilde {\cal  B}}^{ws}_{\hat T}$ of $f$ 
for each $f\in   {\tilde {\cal  B}}^{ws}_{\hat T}$. 

	
 %\proof

 %{The proof of the last statement follows from Lemma [?] in [L?]. The proof of the first statement is in [L?]. The main idea of the proof for the continuity is the following.	First note that we only need to consider the case that
 %	the domains of the $L_k^p$-maps here  are  the fixed one $\Sigma\simeq {\bf CP}^1$ and $G_T=G_i, i=0, 1, 2$ of the standard connected Lie subgroups of ${\bf PSL}(2, {\bf C}).$ We will use  additive notation as if  $\Sigma$  and $M$  have   global coordinate  charts.  Given $f_0\in {\bf B}^{\hat T}$, $g_0\in G_T$ and $\epsilon>0, $  let $f_0'$ be a $C^{\infty}$-map with $\|f_0-f_0'||< \epsilon/(4||g_0||).$
 	%Now $\|f\circ g-f_0\circ g_0\|\leq  \|f\circ g-f'_0\circ g\|+\|f'_0\circ g-f'_0\circ g_0\| $.
 	
 	%To estimate last therm, note that $\|f'_0\circ g-f'_0\circ g_0\|$ is controlled by the following terms with $l+m=k.$
 	
 	 %$$\|(D^lf'_0\circ g)\cdot D^mg -(D^lf'_0\circ g_0)\cdot D^m g_0\|_{0, p}$$ $$ \leq \|((D^lf'_0\circ g)-(D^lf'_0\circ g_0))\cdot D^m g\|_{0, p}+   \|(D^lf'_0\circ g_0)\cdot D^m(g-g_0) \|_{0, p}$$ $$
 	% \leq \|(D^lf'_0\circ g)-(D^lf'_0\circ g_0)\|_{0, p}\cdot\|    g\|_{C^k}+\|D^lf'_0\circ g_0\|_{0, p}\cdot \| g-g_0 \|_{C^k}
 	 % $$ $$ \leq \|f'_0\|_{k+1, p}\cdot  \|g- g_0\|_{C^k}\cdot\|    g\|_{C^k}
 	 %+|| Jac (g_0)\|_{C^k}\cdot \|D^lf'_0\|_{k, p}\cdot \| g-g_0 \|_{C^k}.$$
 	
 	
 %This implies that the last term is less that 	$ \epsilon/4$ for $\|g-g_0\|_{C^k}<\delta$ when $\delta$ is sufficiently small. The first term $\|f\circ g-f'_0\circ g\|<\|Jac (g)\|_{C^k}\|f-f'_0\|<\epsilon/4$. Again here we have assumed that $\delta$ is so small so that $\|g\|_{C^k}$ is almost equal to  $\|g_0\|_{C^k}.$
 	
 	
 	
 	
 %	}

 %\QED


 \section {The properness of $G_T$-action and \\ $G_T$-Hausdorffness of ${\cal B}^{\hat T}$}


  First note that  the following  slightly general  theorem  can be derived from the corresponding main theorem on properness of $G_T$-action in section 1.

 \begin{theorem}
 	The partially defined action of $G_f=:G_{\Sigma_f}$ on  $   {\tilde {\cal  B}}^{ws}_{\hat T}$ is
 	proper in the sense that  for any $f_1$ and $f_2$  $   {\tilde {\cal  B}}^{ws}_{\hat T}$ there exist the corresponding  open neighborhoods $U_1$ and $U_2$ containing $f_1$ and $f_2$ at which
 	$G_{f_1}$ and $G_{f_2}$ act on  respectively  and compact subsets $K_1$ and $K_2$ in 	$G_{f_1}$ and $G_{f_2}$ accordingly   such that  for any $h_1$ in $U_1$ ($h_2$ in $U_2$) and $g_1$ in $G_{f_1}\setminus K_1$  ($g_2$ in $G_{f_2}\setminus K_2$ ), $g_1\cdot h_1$ is not in $U_2$ ($g_2\cdot h_2$ is not in $U_1$).
 	
 	
 	
 \end{theorem}


  Indeed,  if the theorem is not true, there  are  sequences of $h_i\in
  U_1$ with $\lim _{i\rightarrow \infty}h_i=f_1$ and $g_i\in G_{f_1}$ with $g_i$ not staying in any compact set as $i$ goes to infinity such that $g_i\cdot h_i$ is lying in $U_2$. Now
  $g_i=t_i\circ p_i$ with $t_i$ in $G_T$ and $p_i$ in  the finite group
  $G_{\Sigma_{f_1, [{\bf x}]}}.$ Hence after taking a subsequence, we may assume that $p_i$  is a fixed element $p$ in $G_{\Sigma_{f_1, [{\bf x}]}}$  and $t_i$ is not staying  in any compact set of $G_T$ as $i$ goes to infinity.
  Since $G_T$  acts trivially  on the base of the universal curve, the assumption that $g_i\cdot h_i$ is lying in $U_2$ implies that the group $G_{\Sigma_{h_1}, [{\bf x} ]}$  is "covered" by  $G_{\Sigma_{f_2}, [{\bf x} ]}$ so that the action of $p$ on $(\Sigma_{h_1}, [{\bf x}]) $   comes from a corresponding element
  $p'$ in $G_{\Sigma_{f_2, [{\bf x}]}}.$ Hence after conjugating the actions of $p^{-1}$ and  $(p')^{-1}$ accordingly we may assume that
  there are sequences of $h_i\in
  U_1$ with $\lim _{i\rightarrow \infty}h_i=f_1$ and $g_i\in G_T$ with $g_i$ not staying in any compact set as $i$ goes to infinity such that $g_i\cdot h_i$ is lying in $U_2$. This contradicts to the main theorem
  on the properness of $G_T$-action.

  Clearly to prove the main theorem for $G_T$, we only need to look its action on its unstable components. Since $G_T$-action on the domains does not move the  double points, we only need to prove the corresponding statement for a fixed domain $\Sigma\simeq {\bf CP}^1$
  with $G_T=G_i, i=0, 1, 2$ of the subgroups of ${\bf PSL}(2, {\bf C})$ preserving $i$ marked points. The  proofs of the three cases are similar. We only prove the hardest case that  $G_T=G_0= {\bf PSL}(2, {\bf C})=:G$.

  \begin{pro}
  	Let $   {\tilde {\cal  B}}^{*}$ be the Banach manifold of non-trivial $L_k^p$-maps from $\Sigma={\bf CP}^1$ to $M$. Then the  action of  $G={\bf PSL}(2, {\bf C})$ on $   {\tilde {\cal  B}}^{*}$ is proper.
  \end{pro}

  \proof


 We start with some elementary linear algebra.

 For any  $g \in SL(2, {\bf C}),$
 let    $g=h\cdot  u$ be the decomposition in $SL(2, {\bf C})$  with $u\in SU(2)$ and $h$ being self-adjoint.  Here   $h=(g\cdot g^*)^{\frac {1}{2}}\in SL(2, {\bf C})$ and $u= (g\cdot g^*)^{-\frac {1}{2}}\cdot g\in SU(2)$.

 Note that for $g\not \in  SU(2)$ the  decomposition $g\cdot g^*=w^*\cdot diag ( \lambda_1, \lambda_2)\cdot w$ with $w\in SU(2)$ and  $\lambda_1<\lambda_2$ is unique.  So is  $(g\cdot g^*)^{\frac {1}{2}}=w^*\cdot diag ( r_1, r_2)\cdot w.$ Here $r_i=(\lambda_i)^{\frac {1}{2}}>0, i=1, 2$ and $r_1<r_2.$      Rename $w^*$ as $u$ and $wu$ as $v$. Denote $diag ( r_1, r_2)$ by $D({\bf r})$ for short. Then we have the unique decomposition  $g=u\cdot D({\bf r})\cdot v$ in $SL(2)$ with $u$ and $v$ in
 $SU(2)$ for any $g\not \in  SU(2)$.


 %%%%%%%%%%%%%%%%%

  Assume that the proposition  is not true. Then for any small neighbourhoods
  $U_{\epsilon_i}(f_i), i=1, 2 $ and any nested  sequences of compact sets $K_1\subset K_2\subset \cdots \subset K_l\cdots$ in $G$, there  are  sequences
  $\{g_n\}_{n=1}^{\infty}$ in $G$ and $\{h_n\}_{n=1}^{\infty}$ in $U_{\epsilon_1}(f_1)$ such that (a) $g_n$ is not in $K_n$; (b) $h_n\circ
  g_n$ is in  $U_{\epsilon_2}(f_2).$



  Here $\epsilon_i, i=1, 2 $ and   $K_n, n=1, \cdots$,
  will be decided below in the proof.
  \vspace{2mm}
  \noindent

  After taking a subsequence we may assume that $g_n\not \in SU(2)$ so that it has the unique decomposition in ${\bf SL}(2, {\bf C}), $
   $g_n=u_n\cdot D({\bf r}_n)\cdot v_n$ with $u_n$ and $v_n$ in $SU (2)$ and
  $ D({\bf r}_n)=diag (r_{n,1}, r_{n,2})$ with  $0<r_{n,1}< r_{n,2}.$ We may assume that $ r_{n,2}=1$ by considering $ D({\bf r}_n)$ as an element
  in ${\bf PSL}(2, {\bf C}).$
  Denote  $r_{n,1}$  by $a_n$  and
   $ D({\bf r}_n)$ by  $D(a_n)$ or $g_{a_n}$.
   Let ${\cal D}_{2}$ be the collection of all $2\times 2$
  non-singular  diagonal matrices  with positive  entries.

  Let  ${\tilde K}_n\subset  SU (2)\times {\cal D}_{2} \times SU (2)$   be the set of tuples  $(u, D({\bf r}), v)$  with $\frac {1}{n} \leq r_i\leq 1, i=1, 2$, where $D({\bf r})=diag (r_1, 
  r_{2})$.  Denote  the corresponding compact set in  ${\bf PSL}(2, {\bf C})$ by ${ K}_n$.

  % obtained by sending $(u, {\bf r}, v)$ to $u\cdot D({\bf r})\cdot v$  in ${\bf PSL}(2, {\bf C})$.
  % Here  we consider $G$  as a closed subgroup of $PSL(k+1)$, hence  $K_n$
  %is compact.



  In these notations, the condition (a) above implies that for $g_n=u_n\cdot D({a}_n)\cdot v_n$,     $lim_{n\mapsto \infty} a_n=0.$

  After  taking subsequence, we may assume that $lim_{n\mapsto \infty} u_n=u$ and
  $lim_{n\mapsto \infty} v_n=v$  in $SU(2).$







  Let  ${\bf CP}^1={\bf C}\cup {\infty}  $ with complex coordinate of $z\in {\bf C}$.    Denote the   closed   disc  of radius $R$ in ${\bf C}$ centered at origin by   $B(R)$.
  % Denote the complements of $M(R)$ in $M$ and $B(R)$ in ${\bf CP}^k$ by
  %$M^c(R)$ and  $B^c(R)$ respectively. They are small neighbourhoods of $M(\infty)$ in $M$ and $H$ in ${\bf CP}^k$.

  Since $f_2$ is nontrivial, its  energy $E(f_2)=\delta_2>0.$ Then there is a point $x_0\in {\bf CP}^1 $ such at $e(df_2)(x_0)>0.$ We may assume that (i)$x_0\not =\infty$ so that for $R$ large enough,  $x_0 $ is in the interior of $ B(R)$; (ii)    there are   positive constants $\gamma$ and $\rho$ small enough such that the disc  $B(x_0; \rho)$  of radius $\rho$ centered at $x_0$  is in $B(R)$  and that for any
  $x$ in  $B(x_0; \rho)$,  $e(df_2)(x)>\gamma.$ This implies that  for $N_2$ sufficient large, we have $E(f_2|_{B(R)} )\geq E(f_2|_{B(x_0; \rho)} )\geq \delta_2/N_2.$ Clearly  when  ${\tilde \epsilon}_2>0$ is small enough,  for any    $h$ with $\|h-f_2\|_{C^1}<{\tilde  \epsilon}_2$, we still have that $E(h|_{B(R)})\geq \delta_2/N_2.$
  Now $\|h-f_2\|_{C^1}\leq C_2 \cdot \|h-f_2\|_{k, p}$ by our assumption. Hence if we choose $\epsilon_2<{\tilde \epsilon}_2/C_2,$ then for any $h$ in $U_{\epsilon_2}(f_2)$,  $E(h|_{B(R)})\geq \delta_2/N_2.$
  In particular,  since $h_n\circ g_n$ is in $U_{\epsilon_2}(f_2)$, we conclude
  that $E(h_n\circ g_n|_{B(R)})\geq \delta_2/N_2.$ Fix such $R$, $ \delta_2$ and  $N_2$.

  Now under the automorphism  $v $ of $ {\bf CP}^1={\bf C}\cup {\infty}$, the point  $\infty $ maps to $\infty_v:=v(\infty)$,  and  ${\bf C}$ to ${\bf C}_v:=v({\bf C})$, etc. In particular, $B(R)_v=v(B(R)) .$

   Under the identification ${\bf C}_v\simeq {\bf C}$ the standard coordinate $z$ becomes the coordinate   for  ${\bf C}_v$.

  Then in term of this   coordinate  of  ${\bf C}_v$, the action of $D({a_i})$ restricted to ${\bf C}_v$  is given by $D{a_i}(z)= a_i\cdot z$ with  all $a_i\leq 1.$  For any fixed $R>0 $ and any given $\epsilon >0, $   by our assumption,
   when $i$  is large enough, we have $  D({a_i})(B(R)_v)\subset (B(\epsilon)_v)\subset  {\bf C}_v.$  Hence the area $A (D({a_i})(B(R)_{v}))\leq C_3\epsilon^2$.  Here the area is computed with respect to the Fubini-Study metric which is uniformly equivalent
   to the flat metric on $B(R)_v$ for fixed $R$.
  Note that  it follows from the definition of $B(R)_v$,  in  above  inequality, the constant $C_3$ is  independent of $v\in SU(2)$ and $i$ as long as  $i>i_0=i_0(\epsilon).$
  In particular, for  $v=v_i$  with  $i>>i_0$,  the inequality still holds.



  Applying this to $g_i=u_i\circ D({a_i})\circ v_i$, since $u_i$  preserves the  Fubini-Study metric, we conclude that
  for $i$ large enough, $A (g_i(B(R))=A(D({a_i})(B(R)_{v_i}))$ $\leq C_3\epsilon^2.$

  Now $$E  ((h_i\circ g_i)|_{B(R)})=E  (h_i|_{ g_i(B(R))})$$  $$ \leq ||h_i||^2_{C^1} A ( g_i(M(R))\leq C_3\cdot ||h_i||^2_{C^1}\epsilon^2$$
  $$  \leq C_3 ||h_i||^2_{k, p} \epsilon^2  \leq  C_3(\|f_1\|_{k, p}+ ||h_i-f_1||_{k, p})^2\epsilon^2 $$ $$ \leq C_3(\|f_1\|_{k, p}+\epsilon_1)^2 \epsilon^2 . $$

  This implies that  when $\delta_2>0$,  $N_2>0$ (depending on $\|f_2\|_{k, p}$ and $\epsilon_2$ ) and $\epsilon_1$  are  fixed, for   any choice of $\epsilon$,
  when $i>i_0=i_0(\epsilon)$ is large enough, $ \delta_2/N_2<E  ((h_i\circ g_i)|_{B(R)})< C_3(\|f_1\|_{k, p}+\epsilon_1)^2 \epsilon^2 . $  This is impossible.

  QED


 Applying  the main theorem on properness to the case  that $f_1=f_2$,  we get the first  part of the following  corollary.

 \begin{cor}
 	The isotropy group ${\Gamma}_f$ of any weakly stable $L_k^p$-map
 	$f$ is always compact. Moreover the  $G_T$-orbit and hence $G_f$-orbit of $f$  in
 	$   {\tilde {\cal  B}}^{\hat T}$ is closed.
 	

 \end{cor}

 \proof


 We only  need to  prove  the  last statement.
 We only prove this   for the essential case that $f\in  {\tilde {\cal B}}^{ws}_{\hat T}$.
 Rename $f$ as  $f_1$. If the corollary is not true, there exist $g_i\in G$ and $f_2\in {\tilde {\cal B}}^{ws}_{\hat T}$ such that
 $f_2=\lim_{i\mapsto \infty}f_1\circ g_i,$ but $f_2$ is not in $G\cdot f_1$.
 Therefore for any $U_{\epsilon_2}(f_2), $  when $i$ is large enough,  $f_1\circ g_i$ is in $U_{\epsilon_2}(f_2). $
 On the other hand,  the main theorem  with the same notation implies that for all such $i$,  $g_i$ is in the compact set $K_1$. Therefore, we may assume that
 $\lim_{i\mapsto \infty} g_i=g$ in $K_1.$ Consequently, $f_2=\lim_{i\mapsto \infty}f_1\circ g_i=f_1\circ g.$ That is $f_2\in G\cdot f_1$ which is  a contradiction.

 \QED


 Essentially the same argument proves the following stronger result.
 \begin{cor}
 	Given any  non-constant map
 	$f$ in ${\tilde {\cal  B}}^{\hat T}$, there is a small closed ${\delta}$-neighbourhood $B_ {\delta}(f)$ such that the $G_T$-orbit $G\cdot B_ {\delta}(f)$ is closed in ${\tilde {\cal  B}}^{\hat T}$.  In other words, ${\tilde {\cal  B}}^{\hat T}$ is $G_T$-regular in the sense that for any $G$-closed subset $C$ in ${\tilde {\cal  B}}^{\hat T}$  and $f\not \in C$ , there are $G_T$-open neighbourhoods $U_1$ and $U_2$ of $C$ and $G\cdot f$ respectively such that $U_1$ and $U_2$ do not intersect.
 	
 \end{cor}


In fact a similar argument  together  with the fact that any closed and bounded subset of a Banach  space is weakly compact implies the following stronger result. 

\begin{cor}
	Let 
	$C$  be a closed and "bounded " subset in   ${\tilde {\cal  B}}^{\hat T}$, 
	then its $G_T$-orbit $C^{G_T}$ is closed.
	
	
	
\end{cor}
 \proof

Again we only  give the proof  for the essential case that $f\in  {\tilde {\cal B}}^{ws}_{\hat T}$. This can be reduced further by assuming  the domain of the elements in $f\in  {\tilde {\cal B}}^{ws}_{\hat T}$ is the fixed $S^2$. We will assume
 that the target $M$ is $C^{\infty}$ embedded in some ${\bf R}^n $. The compactness of $M$ implies that the induced embedded  ${\tilde {\cal B}}_{\hat T}\rightarrow Map_{k, p}(S^2,{\bf R}^n )$ is closed. Moreover   the topology  given by the Banach structure on ${\tilde {\cal B}}$ is the same as the one induced from the Banach space $Map_{k, p}(S^2,{\bf R}^n ).$  Then the assumption  on  $C$ becomes   the condition that
 there is a closed and bounded subset ${\hat C } $ in $Map_{k, p}(S^2,{\bf R}^n )$ such that  $C={\hat C } \cap Map_{k, p}(S^2,{\bf R}^n ).$ Clearly the $G_T$-orbit
 $C^{G_T}={\hat C}^{G_T}\cap Map_{k, p}(S^2,{\bf R}^n ).$ Hence we only need to prove
 the statement for  $Map_{k, p}(S^2,{\bf R}^n )$.
 
With this understanding, let 
$f_i\in C$ and $g_i\in G$ be two sequences such that
$h=\lim_{i\mapsto \infty}f_i\circ g_i$ in ${\tilde {\cal B}}^{ws}_{\hat T}$ with respect to the $L_k^p$-topology.  The convergence is also with respect to $L_{k-1}^p$-topology. Then for $i$  large enough,  $f_i\circ g_i$ is in $U_{\epsilon_2}(h) $ in $L_{k-1}^p$-topology. On the other hand, 
the assumption on $C$ implies that there exists a $f$ in ${\tilde {\cal B}}^{ws}_{\hat T}$ such that $f=\lim_{i\mapsto \infty}f_i$ in $L_{k-1}^p$-topology so that for $i$  large enough,  $f_i$ is in a $L_{k-1}^p$ neighborhood  $U_{\epsilon_1}(f) $.  Thus the argument before implies that after taking a subsequence, there exists a $g\in G$ such that $g=\lim_{i\mapsto \infty}g_i.$

Denote $ f_i\circ g_i$ by $h_i$. Then $h=\lim_{i\mapsto \infty}h_i$ in $L_k^p$-topology. This implies that $f_i=h_i\circ g_i^{-1}$ is convergent to $h\circ g^{-1}$ in $L_k^p$-topology as well. Clearly the $L_k^p$-limit here is the same as the
 $L_{k-1}^p$-limit $f$ before.  Hence  $h=\lim_{i\mapsto \infty}f_i\circ g_i=f\circ g$
 is in $C^{G_T}.$






\QED

 


 %For  any $U_{\epsilon_2}(f_{ 2}), $  when $j$ is large enough,  $h_j\circ g_{ j}$ is in $U_{\epsilon_2}(f_{2}). $
 %On the other hand,  the Theorem 1.1  implies that  for proper choices   of   the radius ${\delta}$ of  $B_ {\delta}(f_1)$ and the radius ${\epsilon_2}$ of $U_{\epsilon_2}(f_{ 2}) $,    $g_{ j}$ is in the compact set $K_1$ defined in the Theorem 1.1. Therefore, we may assume that
% $\lim_{i\mapsto \infty} g_{ j}=g$ in $K_1.$  Hence
 %$$\lim_{j\mapsto \infty} h_{ j}=\lim_{j\mapsto \infty} (h_{ j}\circ g_{j})\circ  \lim_{j\mapsto \infty} g_{j}^{-1} $$ $$ =f_{ 2}\circ  g^{-1}.$$
% Since $ B_ {\delta_i}(f_1)$  is closed, we have $\lim_{j\mapsto \infty} h_{ j},$ denoted by $h$,  is in  $ B_ {\delta}(f_1)$.
% Therefore, $f_{ 2}=\lim_{j\mapsto \infty}h_{ j}\circ\lim_{j\mapsto \infty} g_{j}=h\circ g\in  G\cdot B_ {\delta}(f_1)$. This is a contradiction.







% \QED


  On the other hand, in the case $f_1\not =f_2$, the conclusion of the main
 theorem is weaker than the corresponding statement on the $G$-Hausdorffness stated in Theorem 1.1.

 Next theorem is a slightly stronger version of the Theorem 1.1

  Given  $f$ in  the space  $   {\tilde {\cal  B}}_{\hat T}$ of  nodal $L_k^p$-maps modeled on ${\hat T}$, let $G_f$ be the group of its reparametrizations. Recall that there is sufficiently small neighborhood $U=U_f$ such that the action $G_f$ extends to
   $U$ such that for  any $h\in U$ the action of $G_f$ covers the action
    of $G_h$ on the  corresponding $U_h\subset U_f$. %In particular, the
    %for any $h\in U$ its $G_h$-orbit is  the same as the  $G_f$-orbit $G_f\cdot h.$  
    Therefore the local model for the quotient space
 $   {{\cal  B}}_{\hat T}$ of  nodal $L_k^p$-maps modeled on $T$ is given by above neighborhood $U_f$ quotient  by $G_f.$

% ${\bullet}$ {\bf  Note:} In the proof above, we only have  used the fact
 % that $f_2$ is not trivial. In the case that $f_1$ is a trivial map.
 % the last inequality becomes $ \delta_2/N_2<E  ((h_i\circ g_i)|_{B(R)})< C_3(\epsilon_1)^2 \epsilon^2  $ when $i>i_0=i_0(\epsilon)$  for  fixed  $\delta_2>0$,  $N_2>0$ (depending on $\|f_2\|_{k, p}$ and $\epsilon_2$ ) and $\epsilon_1$. This leads to a contradiction as  before. This proves part of the following corollary.





  \begin{theorem}
  The space  $   {\tilde {\cal  B}}^{\hat T}$ of  nodal $L_k^p$-maps modeled on ${\hat T}$  is $G_f$-Hausdorff in the sense that for any two diffent $G_f$-orbits $G_{f_1}f_1$ and $G_{f_2}f_2$, there exit $G_f$-neighborhoods $G_{f_1}U_1$ and $G_{f_2}U_2$ such that $G_{f_1}U_1 \cap G_{f_2}U_2=\varphi$. Therefore, the  quotient space $  { {\cal  B}}^{\hat T}$  of unparametrized  nodal $L_k^p$-maps is Hausdroff.
  \end{theorem}

 \proof

 As remarked before, the essential case is the  one  for $G_T$-action. We only prove this case. In this situation, the   proof  reduces
 to consider the two corresponding components defined on the same domain $\Sigma_v\simeq S^2$ with $v\in T$, still denoted by $f_1$ and $f_2$ and the mapping spaces that they lie on.

   The proof consists of three parts.

  ${\bullet}$   Part I:  the case that   both $f_1$ and $f_2$  are  non-trivial.



  By Theorem 1.1, for any $g$ not in the compact set $ K_1$ and $h\in  U_{\epsilon_1}(f_1)$, $h\circ g$ is not in $U_{\epsilon_2}(f_2)$. By our assumption, we may assume that
  $  U_{\epsilon_1}(f_1)$ and $U_{\epsilon_2}(f_2)$ have no intersection.

  \vspace{2mm}
  \noindent
  \vspace{2mm}
  \noindent $\bullet$ Claim: when $\epsilon_i,i=1,2$  are small enough,
  $(G_T\cdot U_{\epsilon_1}(f_1))\cap U_{\epsilon_2}(f_2)$ is empty.

  \proof


  \vspace{2mm}
  \noindent
  If this is not true, there  are $h_i\in U_{\delta_i}(f_1)$ and $g_i\in K_1$
  such that $h_i\circ g_i$ is in $ U_{\delta_i}(f_2)$ with $\delta_i\mapsto 0.$
  The compactness of $K_1$ implies that after taking a subsequence, we have that
  $\lim_{i\mapsto \infty}g_i=g\in K_1.$ Since $\delta_i\mapsto 0$, we have that
  $f_1=\lim_{i\mapsto \infty}h_i$ and $f_2=\lim_{i\mapsto \infty}h_i\circ g_i=f_1\circ g.$
  Hence, $f_1$ and $f_2$ are in the same orbit which contradicts to our assumption.
  Note that in the last identity above, we have used the fact that the action map
  $\Psi: G_T\times   {\tilde {\cal  B}}^{\hat T}\rightarrow   {\tilde {\cal  B}}^{\hat T}$ is continuous.

  \QED

  Of course the same proof also implies that
  $(G_T\cdot U{\epsilon_2}(f_2))\cap U_{\epsilon_1}(f_1)$ is also empty for sufficiently
  small $\epsilon_i, i=1, 2.$


  If $h\in (G_T\cdot  U{\epsilon_1}(f_1))\cap ( G_T\cdot  U_{\epsilon_2}(f_2)), $  then there are
  $h_i\in U_{\epsilon_i}(f_i) $ and $g_i\in G, i=1, 2$ such that
  $h=h_1\circ g_1=h_2\circ g_2. $ Hence $h_2=h_1\circ g_1\circ g_2^{-1}$ and
  $(G_TU_{\epsilon_1}(f_1))\cap U_{\epsilon_2}(f_2)$ is not empty. This contradicts
  to the above claim.

  \QED



 ${\bullet}$   Part II:  the case that   one of  $f_1$   and  $f_2$  is
  trivial  but the other   is not.  Then the desired results follows from the following stronger statement.

  \begin{lemma}
  	Given any two $L_k^p$-maps $f_1$ and $f_2$ with $E(f_1)\not =E(f_2)$, there exit   $G$-neighbourhoods  $  {\bf W}(f_1)$ of $f_1$ and $ {\bf W}(f_2)$ of $f_2$  which  do  not intersect.
  	
  	In particular if $f_1$ is a constant map and $f_2$ is not, then $E(f_1)=0\not =E(f_2)$ and the above conclusion  holds.
  	
  \end{lemma}

  \proof


  Note that the condition $E(f_1)\not =E(f_2)$ implies that $f_1$ and $f_2$ are not in the same $G$-orbit.
  We may assume that $E(f_1)<E(f_2)$. For any $E(f_1)<c<E(f_2), $ since the energy function $E:{\cal M}\rightarrow {\bf R}$ is continuous and $G$-invariant, the inverse images $E^{-1}((-\infty, c))$ and $E^{-1}((c, \infty))$, denoted by  ${\bf W}(f_1)$ and $ {\bf W}(f_2)$,  are two open
  $G$-sets in ${\cal M}$ containing $f_1$ and $f_2$ respectively. Clearly ${\bf W}(f_1)$ and $ {\bf W}(f_2)$ do not intersect.

  \QED


 ${\bullet}$    Part III: the case that   both  $f_1$   and  $f_2$  are trivial.





   Let $B_{\epsilon'_1}(c_1)$ and  $B_{\epsilon'_2}(c_2)$ be two open balls in $M$, which do not intersect. Here $c_1$ and $c_2$ are the values of the two constant maps $f_1$ and $f_2$ respectively. Clearly if $||h_i-f_i||_{C^0}=max_{x\in \Sigma}|h_i(x)-c_i|<\epsilon'_i, i=1, 2, $ then the image of $h_i$ is contained in $B_{\epsilon'_i}(c_i).$ Moreover, since for  any  $h_i$ and $g_i\in G$,
  the  image of $h_i\circ g_i=$ the  image of $h_i,$ for any  $h_1$ and $h_2$ as above, their $G$-orbits
  $G\cdot h_1$ and  $G\cdot h_2$ do not intersect. Clearly by our assumption   for $\epsilon_i<<
  \epsilon'_i,$
  any  $h_i, i=1, 2$ in $U_{\epsilon_i}(f_i)$ satisfies the condition $||h_i-f_i||_{C^0}<\epsilon'_i$, hence $G\cdot U_{\epsilon_1}(f_1)$
  and $G\cdot U_{\epsilon_2}(f_2)$ do not intersect.

  \QED


 \section{  Comparison of finite part of the  reparametrization group $G_{\Sigma}$ with $S_T$}

  Recall that  the reparametrization group $G_{\Sigma}=:G_{\Sigma_f}$
   is generated by its continuous part
  $G_T$ and the finite part $G^0_{\Sigma, [{\bf  x}]}$.

  As mentioned in the introduction, the important subgroups, such as
  isotropy groups in
  $\Gamma_f$ in $G_T$  are  rather special.
 For instance, it follows from the discussion above, for a weakly stable nodal $L_k^p$-map  $f$, the identity component of $\Gamma_f$
  in $G_T$ is a tours $T^n$.

  We now show the finite group $G^0_{\Sigma, [{\bf  x}]}$ itself is  also quite restrictive.

  The finite  group   $G^0_{\Sigma, [{\bf  x}]}$ is isomorphic to $G_{f}/G_T$ that exchanges the components of $f$,  hence induces an injective homomorphism $\Psi: G^0_{\Sigma, [{\bf  x}]}\rightarrow S_T$, where $S_T$ is the group of automorphisms of $T$.
  The above homomorphism is also defined on $G_{\Sigma, [{\bf  x}]}.$   We want to show that the image of $\Psi$ is small so that "most" of
  symmetries of $T$  can not be realized as the automorphisms of $\Sigma.$

  To this end, we start with an elementary discussion on symmetries of $T$. Let $\phi\in S_T$ be an non-trivial symmetry of $T$. Consider the corresponding self map, denoted by $|\phi|:|T|\rightarrow |T|$
  of the underlying space of $T$. It follows from Lefschetz fixed point
  theorem that $\phi$  has at least one  fixed point $x_0\in |T|$.
  Let $F_{|\phi|}$ be the set of fixed points of $|\phi|$.

  The following lemma summaries  important properties of the automorphism $\phi.$

  \begin{lemma}
  (I) 	If  dimension of  $F_{|\phi|}$ is one, then $F_{|\phi|}$ is an subtree of $T$.  For any tip $v$ of $F_{|\phi|}$, let $T_v$ be the subtree consists of vertices that are reachable by chains in $T\setminus  F_{|\phi|}$ starting from $v$. Then the action of $\phi$
  induces an automorphism  of $T_v$ with the only one fixed vertex $v.$

  (II) If the dimension of  $F_{|\phi|}$ is zero, then $|\phi|$ has only one fixed point that is either a vertex $v$ of $T$ or a midpoint $m_0$ of an edge $[v_0v_1]$. In the latter case, $\phi$ is an  involution  with  respect to the midpoint $m_0$.


  (III)
  	The underlying space  $|T|$  can have at most one such midpoint
   with  respect to which there is  an involution $\phi\in S_T$.


  \end{lemma}

  \proof

  The proof  is elementary. We only give the proof for (III).

  {Let $T_0$  ($T_1$ respectively) be the subtree consisting of all vertices that are reachable by a chain  from $v_0$ ($v_1$) without passing $v_1$ ($v_0$). Let $m'_0\not = m_0$ be the midpoint of $[v'_0v'_1]$ for another involution  $\phi'\in S_T.$ Let $T'_0$ and $T'_1$ be the corresponding connecting components. We may assume that $[v'_0v'_1]$ is contained in $T_1$,  and the  distance from $v'_0$ to $v_1$ is one less than  the  distance from $v'_1$ to $v_1.$ Then the chain $[v_1, v'_0]$, the edge $[v_0, v_1]$ and the connecting component $T_0$ is reachable from $v'_0$ without passing $v'_1$.  This implies that
  	$T_0\subset T'_0$, which is impossible unless $T'_i=T_i, i=0, 1$ and
  	$\phi=\phi'.$}

  \QED


  It follows from this lemma that essentially  we only need to consider the case that $|\phi|$ only has one fixed point.

  Consider first  the case that
  there is an involution $\phi\in S_T$ with fixed midpoint $m_0$. Let ${\bf Z}_{2, \phi}$ be the ${\bf Z}_{2}$-subgroup of $S_T$ generated by $\phi.$ The uniqueness of $m_0$ and $\phi$, implies that for any $\phi_1\in S_T$,  $\phi_1^{-1}\circ \phi\circ \phi_1= \phi$  and $\phi_1(m_0)=m_0.$  Hence  in this case, if $\phi_1\not = \phi, $
  the dimension of $ F_{|\phi_1|}$ is one with $m_0\in [v_0, v_1]\subset  F_{|\phi_1|}$.  In other words, once there exists  an involution $\phi\in S_T$ with midpoint $m_0$, all the other
  $\phi_1$ in $S_T$ are in case I above  fixing $[v_0, v_1]$  and commutes with $\phi$. It follows  that in this case, $S_T$ is isomorphic to the direct product of ${\bf Z}_{2, \phi}$ and $S_{T(\phi)}$. Here $T(\phi)$ is the quotient tree   obtained from $T\setminus (v_0, v_1)$  by  the identification of $\phi$.


  Thus with the reductions above,  we only need to consider the essential case that $\phi$ has only one fixed point that is a vertex $v$.


  {\bf Example:}  The simplest case for this is the tree $T(l)$ of $l+1$ vertices, $v_0, v_1, \cdots, v_l,$  with one root $v_0$ and $l$ edges $[v_0 v_1], \cdots, [v_0 v_l]$. Then $S_T$ is the group  $S_l$ of permutations on $l$ letters $v_1, \cdots, v_l$ fixing $v_0$.


  For a general $\phi\in S_T$ with only one fixed vertex $v_0, $
   % Many trees do not have any symmetries.
 % However, if $S_T$ is not trivial, it often contains a subgroup which is isomorphic to the  symmetric group $S_l$ of the permutations on $l$ letters. To explain this,
   % It is easy to see that either $x_0=v_0$ is a vertex or it is a midpoint $m_0$ of an
   %edge $[v_0, v_1]$. In the latter case, let $T_0$ and $T_1$ be the two connected components of   $T\setminus (v_0, v_1)$  connecting $[v_0, v_1]$ at the $v_0$ and $v_1$ respectively.  Then $\phi$ is an involution at identifying $T_1$ and $T_2$. Clearly $\phi$ has no fixed point  on $T$.
  	%Thus upto the possible unique involution without fixed   vertices,
  	% we may assume that any $\phi\in S_T$ has at least one  fixed vertex.  Given $\phi \in S_T, $  fix such  a vertex $v_0.$
  	 let $v_1, v_2,\cdots, v_l$ be the $l$ vertices that are adjacent  to $v_0$. By the assumption,
  	  $\phi$ acts on $\{v_1, v_2,\cdots, v_l\}$  without fixed points.
  	Let $T_1, \cdots, T_l$ be the corresponding subtree   with root $v_k$. More precisely  $T_k, 1\leq k\leq l,$ is defined to be the subtree  of all vertices (and edges) that is reachable from $v_0$  through a chain passing through $v_k$ with $v_k$ as the root (the initial tip).
  	Then  $\phi$ permutes these subtrees. The action of $\phi$ on  $v_1, v_2,\cdots, v_l$ is  decomposed  as a product of independent cyclic cycles.
  First  assume that the action of  $\phi$ itself is a cyclic cycle. This implies that $\phi^k, k=1, \cdots,  l-1$  identifies $T_1$ with $T_k$. By definition $T_k$ is isolated in the sense  that each of its vertices  has no edge relation with any vertices that is not in $T_k$ except the relation $v_0Ev_k$. Let ${T'_k} $ be the tree obtained from $T_k$ by adding the edge $[v_0v_k ]$. Then $T$ is decomposed as an union of these identical trees with a common root $v_0$. Back to the general case, we may assume  that the cyclic cycle decomposition of  $\phi$, considered as an element of $S_l$ by the induced action on $\{v_1, \cdots, v_l\}$,   is maximal comparing to the decompositions of all other elements of $\Gamma_{v_0}$.  Here $\Gamma_{v_0}=\Gamma^T_{v_0}$ be the subgroup of $S_T$ that fixes $v_0$. Indeed, first note that  there is a $\phi_1\in \Gamma_{v_0}$ that
  brings $v_{i'}$ in one cycle $C_i$ to $v_{j'}$ in the other $C_j$ for some $1\leq i\not = j\leq l, $  if and only if there is a   identification of  subtrees
  $T_{i'}$ and $T_{j'}$ with $i'\in C_i$ and $j'\in C_j $. Here  we have used the fact that all $T_k$ are isolated. Then in this case, 
all the subtrees $T_k$  with roots in  $C_i\cup C_j$ are 
  isomorphic.  Again, since all $T_k$ are isolated, this implies that 
  $\Gamma_{v_0}$  contains   an element, still denoted by $\phi, $  whose
 cyclic cycle decomposition contains $C_i\cup C_j$. 
 
 This proves both existence and uniqueness of such maximal $\phi.$ 
Moreover, each cycle $C_i$ associated to $\phi$ is the collection of all isomorphic
subtrees $T_{i'}$ with $i'\in C_i$. Furthermore, 
 the induced action of $\Gamma_{v_0}$ on $\{v_1, \cdots, v_l\}$ preserves each cycle $C_i$. It follows from this that (1)   $\Gamma_{v_0}$ is decomposed as a direct product with each factor associated to a cycle  $C_k$ above and (2)  each factor is  mapping surjectively on to  the symmetry group permutations of the letters in the cycle $C_k$. 
  	
  	 The discussion above implies the following lemma on the structure of each factor.
  	
  	 \begin{lemma}
  	 	 Assume  that  $\phi\in S_T$  has only one fixed vertex $v_0$,  which  has $l$  adjacent vertices that form a cyclic cycle under the permutation by $\phi.$ Then $\phi$ "generates" a subgroup $S^{\phi}_l$ of $\Gamma_{v_0}$ that is isomorphic to $S_l$. Let  $\Gamma^{T_k}_{v_k}$ be the subgroup of $S_{T_k}$ that fixes the root $v_k$, $k=1, \cdots, l.$
  	 	  Then
  	 	 $\Gamma_{v_0}$ contains  $S^{\phi}_l$ as its subgroup such  that $\Gamma_{v_0}$ maps surjectively on $S_l$ so that the exact
  	 	 sequence $1\rightarrow \prod_{k=1}^l\Gamma^{T_k}_{v_k}\rightarrow \Gamma_{v_0}\rightarrow S_l\rightarrow 1$ splits.
  	 	 Consequently $\Gamma_{v_0}$ is a semi-direct product of $\prod_{k=1}^l\Gamma^{T_k}_{v_k}$ and $S^{\phi}_l.$
  	 	\end{lemma}
  	
  	
  	 In general, for a splitting exact sequence $1\rightarrow G_1\rightarrow G_0\rightarrow G_2\rightarrow 1 $, the group $G_0$  as semi direct product  is  defined using  the  conjugation action of $G'_2$
  	 on the normal subgroup $G_1$, where $ G_2'$ is a subgroup of $G_0$ mapping isomorphically to $G_2$ in the above sequence.  For the case at our  hands, the action of $S_l=S^{\phi}_l$ on $\prod_{k=1}^l\Gamma^{T_k}_{v_k}$ is the   one
  	 induced by the conjugation of $\phi$   that permutes the $T_k$.
  	
  	 Thus existence of such maximal  $\phi\in S_T$ with only one fixed vertex $v$ leads to
  	that $\Gamma_v$ is a direct product of the groups, each still denoted by $\Gamma_v$ has   the form  $\Gamma_v =(\prod_{k=1}^l\Gamma^{T_k}_{v_k})\times_{ S_l^{\phi}}S_l$,   a semi direct product induced by the action of $S_l^{\phi}$ on the normal subgroup $\prod_{k=1}^l\Gamma^{T_k}_{v_k}$.
  	
  	 Finally note that in Gromov-Witten theory and Floer homology, there is a principal component in bubble tree so that the corresponding tree has a preferred vertex $v_0$. In this case, $S_T=\Gamma_{v_0}$ by definition.
  	
  	 It  turns out that the  existence of   $(\phi, v_0)$ as above  already implies   that $S_T=\Gamma_{v_0}$. Indeed,  for any $\phi_1\in S_T, $ if ${\hat v}_0=\phi_{1}(v_0)\not= v_0, $ then we may assume that  ${\hat v}_0\in T_1\subset T'_1$. 
  	 %Let
  	 %$C[v_0,{\hat v}_0]$ be  the   chain staring from $v_0$ connecting
  %	$v_0$ and ${\hat v}_0.$  Then  $C[v_0,{\hat v}_0]$
  	 %is also  contained  in $T'_1$.
  	 Denote the image  $\phi_1 (T_k)$ by ${\hat T}_k$ with the root ${\hat v}_k=\phi_1(v_k)$.  Then ${\hat v}_1=\phi_1(v_1)$ is contained in $T'_k$ for some $k$. We claim that $k=1$. Indeed if $k\not =1, $ since $v_0Ev_1$ implies that ${\hat v}_0E{\hat v}_1$, this can happen only when $ {\hat v}_0 =v_1$ and ${\hat v}_1=v_0$. Then $\phi_1$ is an involution on $[v_0, v_1]$. The argument before implies that this is impossible. Thus both $ {\hat v}_0$ ( $\not= v_0$) and $ {\hat v}_1$ are contained in $T'_1$ with $ {\hat v}_1\not = v_0.$ Hence both of them are contained in $T_k$.  Now consider  any vertex $v_{1, i}$ lying on $T_1$ "next"  to $v_1$ with respect to the direction on  edges of $T'_k$
  	 induced by the paths from the root$v_0$ to the tips of $T_1.$  The similar argument implies that ${\hat v_{1, i}}=\phi_1(v_{1, i})$ is still contained in 
  	 $T_1$. Inductively, this implies that $\phi_1$ maps any vertex of $T_1$ into $T_1$.
  	 %Note that each $T_k$ is directed for its edges 
  	 %Then there exits a ${\hat v}_k$,  that is the  vertex lying on $T_1$ next to  ${\hat v}_0$ with respect to the direction of $T_1$.  First assume that all $T_k, k=1, \cdots , l,$ are isomorphic to each other. Then by the construction, the restriction of $\phi_1$ to $T_k$ preserves the direction. This implies that   ${\hat T}'_k$ is contained in $T'_1$. 
  	 This is impossible unless ${\hat v}_0=v_0$.
  	 
  	 
  	 
  	This proves the following lemma.
  	
  \begin{lemma}
  	Assume that  exist a $\phi\in S_T$ with only one fixed vertex $v_0$.
  	Then $S_T=\Gamma_{v_0}.$  Consequently,  $v_0$ is a fixed point of $S_T$.
  \end{lemma}

 Thus the above discussion on $\Gamma_{v_0}$ also gives a inductive "classification" of $S_T$ in above essential case.	 Moreover, the last statement of the above lemma
 suggests that 
 we  can reformulate above classification in term of the fixed point set $F_
  {\Gamma_{v_0}} $ of the whole group $\Gamma_{v_0}$  rather than  the fixed point set  $F_{|\phi|} $ of  a single element.

  Assume that ${\Gamma_{v_0}}\subset S_T$ is non-trivial. Then $v_0\in
  F_{\Gamma_{v_0}}=\cap_{\phi \in \Gamma_{v_0}}  F_{|\phi|} .$
   This implies that $F_{\Gamma_{v_0}}$ is a subtree containing $v_0$.
   Here we use the result before that if a vertex $v_0$ is in the fixed point set  $F_{|\phi|}$, then $F_{|\phi|}$ is a subtree.
  	Then there are two cases: (A) dimension of $F_{\Gamma_{v_0}}$ is zero; (B) dimension of $F_{\Gamma_{v_0}}$ is one. For the case (B),
  for  any tip $v$ of $F_{\Gamma_{v_0}}$, let $T_v$ be the subtree consists of vertices that are reachable by chains in $T\setminus  F_{\Gamma_{v_0}}$ starting from $v$. Then the action of $\Gamma_{v_0}$
  	induces  automorphisms  of $T_v$ with the only one fixed vertex $v$
  	 so that each $T_v$ is in the case (A).
  	
  	  Thus  we only need to consider the case (A). In this case, $F_{\Gamma_{v_0}}=v_0$.   Then there are two subcases:
  	
  	  (A1) There exists
  	  a $\phi \in \Gamma_{v_0}$  such that $v_0$ is the only fixed points.   This case  has been discussed throughly above with an inductive classification of $\Gamma_{v_0}$ which is equal to $S_T$.
  	
  	 (A2)  For all $\phi \in \Gamma_{v_0}, $  dimension of $F_{|\phi|}$ is always one. We claim that  this can not occur. Indeed, if this does happen, consider the vertices $v_1, \cdots, v_l$ adjacent to $v_0$. Since $v_0$ is the only common fixed point, for any $v_k, 1\leq k\leq l, $ there exists a $\phi_{jk}$ such that $\phi_{jk}(v_k)=v_j, 1\leq j\leq l$  and $j\not =k.$ Otherwise
  	 $[v_0v_k]$ is a common fixed edge for all $\phi \in \Gamma_{v_0}. $
  	 Hence the subtree $T_k$ and $T_j$ are identical. Continue this fashion, we conclude that the $l$ subtrees $T_1, \cdots T_l$ are divided into independent  cyclic cycles of length at least 2 such that for  each cycle of length $l_i$ there is a $S_{l_i}$ subgroup
  	 contained in $\Gamma_{v_0} $ and permutes the corresponding trees over the cycle.  Clearly there exists a $\phi\in  \Gamma_{v_0}$ whose induced action on $\{v_1,\cdots, v_l\}$ is the product
  	 of these cycles. Then $v_0$ is the only fixed point for $\phi.$
  	 
  	 
  	 Combining with the  results before, this proves the following lemma.
  	
  	 \begin{lemma}
  	 The common  fixed point set  $F_{\Gamma_{v_0}}$ is the single point $v_0$ if and only if  there is a $\phi \in \Gamma_{v_0}$  whose fixed point set  $F_{|\phi|} =v_0$. Moreover, in this case, $S_T=\Gamma_{v_0}.$	
  	 \end{lemma}
  	
  	 Thus case (A) here is equivalent to the essential case before.
  	
  	 Now assume that $\phi\in S_T$  with the property in the above lemma  is obtained as an  induced automorphism on $T$ from an automorphism  ${\hat \phi}\in G_{\Sigma, [{\bf x}]}.$ Then $\Sigma$
  	 is decomposed as  $l$ identical  bubble trees $\Sigma_{T'_1}, \cdots, \Sigma_{T'_l}$ with the common root component ${\bf CP}^1_{v_0}$.
  	
  	 The subgroup ${\hat \Gamma}_{v_0}$ corresponding to  $\Gamma_{v_0}$
  	  above  consists of  automorphisms in $G_{\Sigma, [{\bf x}]}$ that preserve the component ${\bf CP}^1_{v_0}$,  which induces a homomorphism from ${\hat \Gamma}_{v_0}$  into ${\bf PSL}(2, {\bf C})$. Of course the image of the  homomorphism  here contains the cyclic group $C_l$ generated by the image of $\phi$.
  	
  	
  	
  	
  	
  	
  	 The next   well-known proposition shows that instead of  having $S_l$ as its image,  when $l>6$, the  largest possibility for the image of  ${\hat \Gamma}_{v_0}$ in  ${\bf PSL}(2, {\bf C})$ is the dihedral group $D_l$. In other words, most of the
  	 permutations in $S_l$ can not be realized as the symmetries of $\Sigma$ despite of the fact that $\Sigma$ is decomposed as  $l$ identical  bubble trees with a common root component. Roughly speaking,  inductively, this show that in  each basic factor $S_l$ in the "structure theorem"
  	 for $S_T$, only $C_l$ or $D_l$ is realizable as the corresponding  symmetries of $\Sigma$.
  	
  	 \begin{pro}
  	 Any finite subgroup of ${\bf PSL}(2, {\bf C})$  is  isomorphic to ( and conjugate to)  one of the  the following groups:
  	   (1) the standard cyclic	group $C_l$; (2)  the standard  dihedral group $D_l$; (3) the rotational symmetry  group of a regular polyhedron in ${\bf R}^3$.
  	 \end{pro}
  	
Note that in  the exceptional case (3), the possible finite groups are $A_4$, $S_4$ or $A_5.$ In particular, non of these groups contains an element of order greater than six.

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%D. Henderson,  Stable classification of  Infinite-dimensional manifolds,
%{\it Invetiones math.} {\bf 12}, 48-56  (1971).

%
%
%\bibitem[F]{f}
%A. Floer, Symplectic fixed points and holomorphic spheres,
%{\it  Commun. Math. Phys. } {\bf 120,} 575-611  (1989).

%


%
%\bibitem[H3]{h3}
%D. Henderson,  Corrections and  extensions of  two papers about   infinite-dimensional manifolds,
%{\it General topology and its applications 1}, 321-327  (1971).

%
%
%\bibitem[H]{h}
% H. Hofer,
%A General Fredholm Theory and Applications,
%{\it Preprint} (2005), arXiv: math.0509366[math.SG].
%


%
%\bibitem[HWZ]{HWZ}
%H. Hofer, Wysocki and Zehnder,
%{\it  Preprint} (2011), arXiv: math.11072097[math.SG].
%} 1972.
%
%

%
\bibitem[L1]{l1}
G. Liu,    Weakly Smoothness in GW Theory,
{\it  Preprint} (2013 ).

%

%
%\bibitem[L2]{l2}
%G. Liu,  Smoothness of the Moduli Space of stable $J$-holomorphic Maps,
%{\it In Preparation}.

%
%

%




%
%\bibitem[L4]{l4}
%G. Liu,    Smooth  Structures on Moduli Spaces of Floer trajectories,
%{\it In Preparation}.
%


%
%\bibitem[L5]{l5}
%G. Liu,   On Foundation of Gromov-Witten theory ,
%{\it In Preparation}.
%









%\bibitem[L6]{l6}
%G. Liu,    Sc-actions on Sobolev Manifolds,
%{\it J. In Preparation}.

%
%

%

%
%\bibitem[W]{w}
% K. Wehrheim,  Fredholm Notions in Scale calculus and Hamiltonian Floer Theory,
%{\it  Preprint} (2012), arXiv: math.12094040[math.SG].
%

%

\end{thebibliography}

\end{document}



1509.07187.extracted_bib
[H3]{h3 %D. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, %{General topology and its applications 1, 321-327 (1971). % %
1509.07187.extracted_bib
[La]{La S. Lang, Differential Manifolds, {Springer-Verlag 1972. % % % %
1509.07187.extracted_bib
[H2]{h2 %D. Henderson, Stable classification of Infinite-dimensional manifolds, %{Invetiones math. {12, 48-56 (1971). % % %
1509.07187.extracted_bib
[F]{f %A. Floer, Symplectic fixed points and holomorphic spheres, %{Commun. Math. Phys. {120, 575-611 (1989). % % %
1509.07187.extracted_bib
[H3]{h3 %D. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, %{General topology and its applications 1, 321-327 (1971). % % %
1509.07187.extracted_bib
[H]{h % H. Hofer, % A General Fredholm Theory and Applications, % {Preprint (2005), arXiv: math.0509366[math.SG]. % %
1509.07187.extracted_bib
[HWZ]{HWZ H. Hofer, Wysocki and Zehnder, Applications of polyfold theo ry I: Gromov-Witten theory, {Preprint (2011), arXiv: math.11072097[math.SG]. % % %
1509.07187.extracted_bib
[L]{l G. Liu, Weakly Smoothness in GW Theory, {Preprint (2013 ) arXiv:1310.7209v9 [math.SG]. % %
1509.07187.extracted_bib
[L1]{l1 G. Liu, On the Action of Reparametrization Group on the Space of $L_k^p$-maps I, {Preprint(2013). arXiv:1312.4222v3 [math.SG]. % %
1509.07187.extracted_bib
[L2]{l2 G. Liu, $C^{m_0$-Smoothness of Evaluation Maps, {Preprint(2013) arXiv:1312.2161v1 [math.SG]. % % % %
1509.07187.extracted_bib
[LT]{lt %G. Liu and G. Tian, Floer homology and Arnold cojecture, %{J. Diff. Geom. {49 No.2 (1998). % % %
1509.07187.extracted_bib
[McW]{mcw4 % D. McDuff and K. Wehrheim, Smooth Kuranishi Structures with trivial isotropy, % {Preprint 2012. % % %
1509.07187.extracted_bib
[L5]{l5 %G. Liu, On Foundation of Gromov-Witten theory , %{In Preparation. % %
1509.07187.extracted_bib
[L6]{l6 %G. Liu, Sc-actions on Sobolev Manifolds, %{J. In Preparation. % % % %
1509.07187.extracted_bib
[MS]{ms D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, {AMS 2012. % %
1509.07187.extracted_bib
[H2]{h2 %D. Henderson, Stable classification of Infinite-dimensional manifolds, %{Invetiones math. {12, 48-56 (1971). % % %
1509.07187.extracted_bib
[F]{f %A. Floer, Symplectic fixed points and holomorphic spheres, %{Commun. Math. Phys. {120, 575-611 (1989). % % %
1509.07187.extracted_bib
[H3]{h3 %D. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, %{General topology and its applications 1, 321-327 (1971). % % %
1509.07187.extracted_bib
[H]{h % H. Hofer, %A General Fredholm Theory and Applications, %{Preprint (2005), arXiv: math.0509366[math.SG]. % % %
1509.07187.extracted_bib
[HWZ]{HWZ %H. Hofer, Wysocki and Zehnder, %{Preprint (2011), arXiv: math.11072097[math.SG]. % 1972. % % %
1509.07187.extracted_bib
[L1]{l1 G. Liu, Weakly Smoothness in GW Theory, {Preprint (2013 ). % % %
1509.07187.extracted_bib
[L2]{l2 %G. Liu, Smoothness of the Moduli Space of stable $J$-holomorphic Maps, %{In Preparation. % % % % %
1509.07187.extracted_bib
[L4]{l4 %G. Liu, Smooth Structures on Moduli Spaces of Floer trajectories, %{In Preparation. % % %
1509.07187.extracted_bib
[L5]{l5 %G. Liu, On Foundation of Gromov-Witten theory , %{In Preparation. % %
1509.07187.extracted_bib
[L6]{l6 %G. Liu, Sc-actions on Sobolev Manifolds, %{J. In Preparation. % % % % %
1509.07187.extracted_bib
[W]{w % K. Wehrheim, Fredholm Notions in Scale calculus and Hamiltonian Floer Theory, %{Preprint (2012), arXiv: math.12094040[math.SG]. % %