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 in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups.

中文翻译：

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图产品和相对双曲线群中的计数问题

我们研究双曲空间等距群的一般元素的性质。在一般组合条件下，我们证明 loxodromic 元素是通用的（即，它们在 Cayley 图中的单词度量的球数方面具有完整的密度）并且翻译长度线性增长。我们提供对一大类相对双曲线群和图形产品的应用，包括所有直角Artin群和直角Coxeter群。