Volume 16, issue 2 (2012)

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Pattern rigidity and the Hilbert–Smith conjecture

Mahan Mj

Geometry & Topology 16 (2012) 1205–1246

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.

  1. If Qu 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 Qu.
  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, is the collection of translates of L under G, and Q is any pattern-preserving group of homeomorphisms of G preserving 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.

metric measure space, hyperbolic group, homology manifold, conformal dimension, codimension one subgroup, Hilbert–Smith conjecture, pattern rigidity, Poincaré duality group
Mathematical Subject Classification 2010
Primary: 20F67
Secondary: 57M50, 22E40
Received: 9 January 2010
Revised: 24 March 2012
Accepted: 15 April 2012
Published: 10 July 2012
Proposed: Benson Farb
Seconded: David Gabai, Ronald Fintushel
Mahan Mj
School of Mathematical Sciences
RKM Vivekananda University
PO Belur Math
Dt Howrah WB-711202