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Abstract
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Let
be a proper CAT(0)
space and let
be a cocompact
group of isometries of
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
-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
-hyperbolic group. In
particular the action of
on
is
minimal if
is not virtually cyclic. We also establish a uniform convergence result that is similar to the
-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
to be
-hyperbolic.
We prove that if the contracting boundary is compact, locally compact or metrizable,
then
is
-hyperbolic.
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Keywords
geometric group theory, Morse boundary, rank-one
isometries, CAT(0) geometry, contracting boundary
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Mathematical Subject Classification 2010
Primary: 20F65
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Milestones
Received: 5 July 2016
Revised: 1 February 2018
Accepted: 23 March 2018
Published: 18 April 2019
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