Abstract
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.
Citation
Mahan Mj. "Pattern rigidity and the Hilbert–Smith conjecture." Geom. Topol. 16 (2) 1205 - 1246, 2012. https://doi.org/10.2140/gt.2012.16.1205
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