Pattern Rigidity and the Hilbert-Smith Conjecture

Mahan Mj

doi:10.2140/gt.2012.16.1205

paper

Pattern Rigidity and the Hilbert-Smith Conjecture

arXiv:0906.4243 · doi:10.2140/gt.2012.16.1205

Abstract

In this paper we initiate a study of the topological group $P P Q I (G, H)$ 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 $d i m_{H} < d i m_{t} + 2$ , where $d i m_{H}$ is the Hausdorff dimension and $d i m_{t}$ 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.

32 pages, no figures: final version to appear in Geometry and Topology

Pattern Rigidity and the Hilbert-Smith Conjecture

Abstract

References in corpus (3)

Cited by in corpus (7)