Pattern rigidity and the Hilbert–Smith conjecture

Abstract

We initiate a study of the topological group PPQI(G,H) of pattern-preserving quasi-isometries for G a hyperbolic Poincaré duality group and H an infinite quasiconvex subgroup of infinite index in G. Suppose ∂G admits a visual metric d with dimhaus < dimt + 2, where dimhaus is the Hausdorff dimension and dimt is the topological dimension of (∂G,d). Equivalently suppose that ACD(∂G) < dimt + 2, where ACD(∂G) denotes the Ahlfors regular conformal dimension of ∂G. 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. 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, ℒ is the collection of translates of L under G, and Q is any pattern-preserving group of homeomorphisms of ∂G preserving 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.