Contracting elements and conjugacy growth in Coxeter groups, graph products, and further groups
Laura Ciobanu, Anthony Genevois
TL;DR
This work constructs contracting elements in the standard Cayley graphs of periagroups, including Coxeter groups, graph products, and Dyer groups, to establish acylindrical hyperbolicity unless the group virtually splits as a product. The authors develop a novel geometric framework based on paraclique graphs and their quasi-median closures, together with systems of coherent local metrics, to recognize contracting elements via well-separated hyperplanes. These contracting elements yield sharp asymptotics for conjugacy growth, implying that the conjugacy growth series of non-virtually-abelian instances are transcendental; direct-product decompositions further refine when transcendence holds. The results provide a unified approach to contracting phenomena across periagroups and their subfamilies, with explicit criteria for when acylindrical hyperbolicity and contracting elements occur, and detailed implications for conjugacy growth and Morse elements.
Abstract
In this article we construct contracting elements in the standard Cayley graphs of the so-called periagroups, a family of groups introduced by the second-named author which include Coxeter groups, graph products, and Dyer groups. As a consequence, we deduce that, unless they virtually split as direct products, periagroups are acylindrically hyperbolic and their conjugacy growth series, with respect to standard generating sets, are transcendental.