TOPOLOGY AND DYNAMICS OF THE CONTRACTING BOUNDARY OF COCOMPACT CAT(0) SPACES

摘要

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 δ-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 δ- hyperbolic group. In particular the action of G on ?_cX is minimal if G is not virtually cyclic. We also establish a uniform convergence result that is similar to the π-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 δ-hyperbolic. We prove that if the contracting boundary is compact, locally compact or metrizable, then G is δ-hyperbolic.