Counting automorphic orbits in finitely generated groups

TL;DR

This paper introduces and analyzes the automorphic growth function in finitely generated groups, classifying its behavior in various groups and showing that certain groups have exponential conjugacy growth.

Contribution

It provides a classification of automorphic growth rates for specific classes of groups and demonstrates that Thompson's groups have exponential conjugacy growth.

Findings

Automorphic growth is not a commensurability invariant.

Virtually abelian groups of rank ≤ 2 have classified automorphic growth rates.

Thompson's groups T and V exhibit exponential conjugacy growth.

Abstract

We study an analogue of the conjugacy growth function in finitely generated groups: the automorphic growth function. This counts the number of automorphic orbits that intersect the ball of radius $n$ in the group. We show that this is not a commensurability invariant, by giving virtually abelian counterexamples. We classify the automorphic growth rate of all virtually abelian groups of rank at most $2$, the Heisenberg group, finite rank free groups and Thompson's groups $T$ and $V$. This last computation allows to conclude that $T$ and $V$ have exponential conjugacy growth.