Systolic growth of linear groups

TL;DR

This paper proves that finitely generated linear groups have at most exponential residual girth, and if not virtually nilpotent, this residual girth is exactly exponential, providing insights into their finite quotients.

Contribution

It establishes exponential bounds on the residual girth of finitely generated linear groups, extending understanding of their finite quotients and growth properties.

Findings

Residual girth of finitely generated linear groups is at most exponential

If the group is not virtually nilpotent, residual girth is exactly exponential

Provides bounds on the size of finite quotients for linear groups

Abstract

We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually nilpotent, it follows that the residual girth is precisely exponential.