Topology and Dynamics of the Contracting Boundary of Cocompact CAT(0) Spaces

TL;DR

This paper studies the contracting boundary of proper CAT(0) spaces with cocompact isometry groups, establishing its similarities to Gromov boundaries, analyzing its topological properties, and characterizing when the group is hyperbolic.

Contribution

It provides a detailed comparison between the contracting boundary and Gromov boundary, proves new dynamics and convergence results, and characterizes hyperbolicity via boundary properties.

Findings

The action of G on the contracting boundary is minimal if G is not virtually cyclic.

Established a uniform convergence result similar to $ ext{π}$-convergence.

Proved that compactness, local compactness, or metrizability of the boundary implies G is hyperbolic.

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 $\partial_cX$ 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.