On conjugacy growth of linear groups

TL;DR

This paper proves that finitely generated non-virtually-solvable linear groups exhibit uniform exponential conjugacy growth, with the number of distinct characteristic polynomials growing exponentially, depending only on the dimension.

Contribution

It establishes uniform exponential conjugacy growth for non-virtually-solvable linear groups and quantifies the growth rate based solely on the dimension.

Findings

Finitely generated non-virtually-solvable subgroups of GL_d have exponential conjugacy growth.

Number of distinct characteristic polynomials grows exponentially with the word radius.

Growth rate is bounded away from zero depending only on the dimension d.

Abstract

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.