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

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.