Vol. 299, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 301: 1
Vol. 300: 1  2
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Online Archive
The Journal
Editorial Board
Special Issues
Submission Guidelines
Submission Form
Author Index
To Appear
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Other MSP Journals
This article is available for purchase or by subscription. See below.
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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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