  In this paper we initiate a study of the topological group $PPQI(G,H)$ of
pattern-preserving quasi-isometries for $G$ a hyperbolic Poincare duality group
and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose
$\partial G$ admits a visual metric $d$ with $dim_H < dim_t +2$, where $dim_H$
is the Hausdorff dimension and $dim_t$ is the topological dimension of
$(\partial G,d)$. a) If $Q_u$ is a group of pattern-preserving uniform
quasi-isometries (or more generally any locally compact group of
pattern-preserving quasi-isometries) containing $G$, then $G$ is of finite
index in $Q_u$. b) If instead, $H$ is a codimension one filling subgroup, and
$Q$ is any group of pattern-preserving quasi-isometries containing $G$, then
$G$ is of finite index in $Q$. Moreover, (Topological Pattern Rigidity) if $L$
is the limit set of $H$, $\LL$ is the collection of translates of $L$ under
$G$, and $Q$ is any pattern-preserving group of {\it homeomorphisms} of
$\partial G$ preserving $\LL$ and containing $G$, then the index of $G$ in $Q$
is finite. We find analogous results in the realm of relative hyperbolicity,
regarding an equivariant collection of horoballs as a symmetric pattern in a
{\it hyperbolic} (not relatively hyperbolic) space. Combining our main result
with a theorem of Mosher-Sageev-Whyte, we obtain QI rigidity results. An
important ingredient of the proof is a version of the Hilbert-Smith conjecture
for certain metric measure spaces, which uses the full strength of Yang's
theorem on actions of the p-adic integers on homology manifolds. This might be
of independent interest.
