We initiate a study of the topological group
of pattern-preserving
quasi-isometries for a hyperbolic
Poincaré duality group and
an infinite quasiconvex subgroup of infinite index in
. Suppose
admits a
visual metric
with , where
is the Hausdorff
dimension and is the
topological dimension of .
Equivalently suppose that ,
where
denotes the Ahlfors regular conformal dimension of
.
If
is a group of pattern-preserving uniform quasi-isometries (or more generally
any locally compact group of pattern-preserving quasi-isometries) containing
,
then
is of finite index in .
If instead,
is a codimension one filling subgroup, and
is any group of pattern-preserving quasi-isometries containing ,
then
is of finite index in .
Moreover, if
is the limit set of ,
is the collection of translates of
under ,
and
is any pattern-preserving group of homeomorphisms of
preserving
and containing ,
then the index of
in
is finite (Topological Pattern Rigidity).
We find analogous results in the realm of relative hyperbolicity, regarding an equivariant
collection of horoballs as a symmetric pattern in the universal cover of a complete
finite volume noncompact manifold of pinched negative curvature. Our main result
combined with a theorem of Mosher, Sageev and Whyte gives 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.
Keywords
metric measure space, hyperbolic group, homology manifold,
conformal dimension, codimension one subgroup,
Hilbert–Smith conjecture, pattern rigidity, Poincaré
duality group
