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

TL;DR

This paper characterizes contracting isometries in CAT(0) cube complexes, introduces a combinatorial boundary, and applies these concepts to diagram groups to determine conditions for acylindrical hyperbolicity.

Contribution

It provides a new characterization of contracting isometries without local finiteness assumptions and links these to the boundary structure in diagram groups, leading to criteria for acylindrical hyperbolicity.

Findings

Contracting isometries are characterized without local finiteness assumptions.

A combinatorial boundary for CAT(0) cube complexes is introduced.

Criteria for when diagram groups are acylindrically hyperbolic are established.

Abstract

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 case, we are able to deduce a criterion to determine precisely when a diagram group is acylindrically hyperbolic.