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 ${\partial }_{c}X$ 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.

Fixed some minor errors. 25 pages, 7 figures

References in corpus (1)

• Hyperbolicity via Geodesic Stability

Cited by in corpus (9)

• Stability and the Morse boundary
• Quasi-Mobius Homeomorphisms of Morse boundaries
• Comparing topologies on the Morse boundary and quasi-isometry invariance
• Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization
• A Symbolic coding of the Morse boundary
• A survey on Morse boundaries & stability
• A rank-one CAT(0) group is determined by its Morse boundary
• An embedding of the Morse boundary in the Martin boundary
• Connected components of Morse boundaries of graphs of groups