Orders of elements in finite quotients of Kleinian groups

TL;DR

This paper proves that in most cases, elements of finitely generated, torsion-free Kleinian groups can be mapped to elements of finite groups with a specified order, demonstrating a broad flexibility in their finite quotients.

Contribution

It establishes that all but finitely many elements in such groups admit a given integer as a finitistic order, extending understanding of their finite quotients.

Findings

Almost all elements admit a given order as a finitistic order

Finitistic orders exist for all but finitely many elements

Results apply to finitely generated, torsion-free Kleinian groups

Abstract

A positive integer $m$ will be called a {\it finitistic order} for an element $\gamma$ of a group $\Gamma$ if there exist a finite group $G$ and a homomorphism $h:\Gamma\to G$ such that $h(\gamma)$ has order $m$ in $G$. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian group admit a given integer $m>2$ as a finitistic order.