Mj, Mahan
(2012)
*Pattern rigidity and the Hilbert–Smith conjecture*
Geometry & Topology Monographs, 16
(2).
pp. 1205-1246.
ISSN 1464-8997

Full text not available from this repository.

Official URL: http://www.msp.warwick.ac.uk/gt/2012/16-02/gt-2012...

Related URL: http://dx.doi.org/10.2140/gt.2012.16.1205

## 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.

Item Type: | Article |
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Source: | Copyright of this article belongs to Geometry and Topology Publications. |

Keywords: | Metric Measure Space; Hyperbolic Group; Homology Manifold; Conformal Dimension; Codimension One Subgroup; Hilbert–Smith Conjecture; Pattern Rigidity; Poincaré Duality Group |

ID Code: | 95718 |

Deposited On: | 26 Nov 2012 10:04 |

Last Modified: | 26 Nov 2012 10:04 |

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