Vol. 299, No. 1, 2019

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

Devin Murray

Vol. 299 (2019), No. 1, 89–116

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.

geometric group theory, Morse boundary, rank-one isometries, CAT(0) geometry, contracting boundary
Mathematical Subject Classification 2010
Primary: 20F65
Received: 5 July 2016
Revised: 1 February 2018
Accepted: 23 March 2018
Published: 18 April 2019
Devin Murray
Department of Mathematics
Brandeis University
Waltham, MA
United States