A general construction of simultaneously hyperbolic elements

TL;DR

This paper provides a broad construction method for simultaneously hyperbolic elements in groups acting on hyperbolic spaces, demonstrating their positive density and generalizing several prior results in geometric group theory.

Contribution

It introduces a general construction for simultaneously hyperbolic elements under minimal conditions, extending previous work and establishing their positive density in groups.

Findings

Simultaneously hyperbolic elements have positive density in groups.

The set of simultaneously contracting elements also has positive density.

The results generalize multiple classical counting theorems.

Abstract

In this paper, we give an explicit construction of simultaneously hyperbolic elements in a group acting on finitely many Gromov-hyperbolic spaces under the weakest conditions. This essentially generalizes results of Clay-Uyanik in \cite{CU18}, of Genevois in \cite{Gen19}, and of Balasubramanya-Fern\'{o}s in \cite{BF24}. Besides, we show that the set of simultaneously hyperbolic elements has strictly positive density with respect to any proper word metric under the weakest conditions. This recovers many classical counting results, eg. the main result of Wiest in \cite{Wie17}. As an important ingredient in the proof of main results, we show that the set of simultaneously contracting elements in a group acting on finitely many metric spaces with contracting property has strictly positive density with respect to any proper word metric. This generalizes two results of Wan-Xu-Yang in \cite{WXY24} and of Balasubramanya-Fern\'{o}s in \cite{BF24}.