Topology and dynamics of the contracting boundary of cocompact CAT(0) spaces

Devin Murray

Vol. 299 (2019), No. 1, 89–116

DOI: 10.2140/pjm.2019.299.89

Abstract

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $\delta$-hyperbolic spaces. We will make this comparison more precise by establishing some well-known results for the Gromov boundary in the case of the contracting boundary. We show that the dynamics on the contracting boundary is very similar to that of a $\delta$-hyperbolic group. In particular the action of $G$ on ${\partial }_{c}X$ is minimal if $G$ is not virtually cyclic. We also establish a uniform convergence result that is similar to the $\pi$-convergence of Papasoglu and Swenson and as a consequence we obtain a new North-South dynamics result on the contracting boundary. We additionally investigate the topological properties of the contracting boundary and we find necessary and sufficient conditions for $G$ to be $\delta$-hyperbolic. We prove that if the contracting boundary is compact, locally compact or metrizable, then $G$ is $\delta$-hyperbolic.

Keywords

geometric group theory, Morse boundary, rank-one isometries, CAT(0) geometry, contracting boundary

Mathematical Subject Classification 2010

Primary: 20F65

Milestones

Received: 5 July 2016

Revised: 1 February 2018

Accepted: 23 March 2018

Published: 18 April 2019