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