On conjugacy growth for solvable groups

TL;DR

This paper proves that finitely generated solvable groups that are not virtually nilpotent exhibit exponential growth in the number of conjugacy classes as their elements grow larger.

Contribution

It establishes a clear dichotomy in conjugacy growth behavior for solvable groups based on their nilpotency properties.

Findings

Finitely generated solvable non-virtually nilpotent groups have exponential conjugacy growth.

Virtually nilpotent solvable groups do not necessarily have exponential conjugacy growth.

Provides a classification criterion based on conjugacy growth for solvable groups.

Abstract

We prove that a finitely generated solvable group which is not virtually nilpotent has exponential conjugacy growth.