Pattern rigidity and the Hilbert–Smith conjecture

Mj, Mahan¹

¹ School of Mathematical Sciences, RKM Vivekananda University, PO Belur Math, Dt Howrah WB-711202, India

Geometry & topology, Tome 16 (2012) no. 2, pp. 1205-1246.

Résumé

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 $\partial G$.

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.

Détail

DOI : 10.2140/gt.2012.16.1205

Classification : 20F67, 57M50, 22E40
Keywords: metric measure space, hyperbolic group, homology manifold, conformal dimension, codimension one subgroup, Hilbert–Smith conjecture, pattern rigidity, Poincaré duality group

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