Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups

Genevois, Anthony¹

¹Université Paris-Sud, Orsay, France

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 49-134

Résumé

The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) 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.

Détail

DOI : 10.2140/agt.2020.20.49

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

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