Volume 20, issue 1 (2020)

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Contracting isometries of $\mathrm{CAT}(0)$ cube complexes and acylindrical hyperbolicity of diagram groups

Anthony Genevois

Algebraic & Geometric Topology 20 (2020) 49–134
Abstract

The main technical result of this paper is to characterize the contracting isometries of a $CAT\left(0\right)$ cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a $CAT\left(0\right)$ cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir’s diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated Farley cube complex. As a consequence, in some specific cases, we are able to deduce a criterion to determine precisely when a diagram group is acylindrically hyperbolic.

Keywords
diagram groups, $\mathrm{CAT}(0)$ cube complexes, acylindrically hyperbolic groups
Mathematical Subject Classification 2010
Primary: 20F65, 20F67