The automorphic growth function is not a commensurability invariant, with explicit classifications showing exponential conjugacy growth for Thompson's groups T and V.
Olukoya,The core growth of strongly synchronizing transducers, arxiv e-printsarxiv:2004.00516(2020)
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Counting automorphic orbits in finitely generated groups
The automorphic growth function is not a commensurability invariant, with explicit classifications showing exponential conjugacy growth for Thompson's groups T and V.