Pattern rigidity and the Hilbert–Smith conjecture

Abstract

We initiate a study of the topological group $PPQI\left(G,H\right)$ of pattern-preserving quasi-isometries for $G$ a hyperbolic Poincaré duality group and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose $\partial G$ admits a visual metric $d$ with ${dim}_{haus}<{dim}_t+2$, where ${dim}_{haus}$ is the Hausdorff dimension and ${dim}_t$ is the topological dimension of $\left(\partial G,d\right)$. Equivalently suppose that $ACD\left(\partial G\right)<{dim}_t+2$, where $ACD\left(\partial G\right)$ denotes the Ahlfors regular conformal dimension of $\partial G$.

1. 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 $Q_u$.

2. 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, if $L$ is the limit set of $H$, $\mathcal{ℒ}$ is the collection of translates of $L$ under $G$, and $Q$ is any pattern-preserving group of homeomorphisms of $\partial G$ preserving $\mathcal{ℒ}$ and containing $G$, then the index of $G$ in $Q$ 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.