Counting problems in graph products and relatively hyperbolic groups

TL;DR

This paper investigates the prevalence of loxodromic elements in groups acting on hyperbolic spaces, demonstrating their genericity and linear growth of translation length, with applications to various hyperbolic and graph product groups.

Contribution

It establishes genericity of loxodromic elements under broad conditions and applies results to relatively hyperbolic groups and graph products.

Findings

Loxodromic elements are generic in the studied groups.

Translation length of generic elements grows linearly.

Results apply to right-angled Artin and Coxeter groups.

Abstract

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.