Counting automorphic orbits in finitely generated groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918476779618304 |
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| author | Elliott, Luna Evetts, Alex Levine, Alex |
| author_facet | Elliott, Luna Evetts, Alex Levine, Alex |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18104 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting automorphic orbits in finitely generated groups Elliott, Luna Evetts, Alex Levine, Alex Group Theory 20F69, 20F65, 20F28 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. |
| title | Counting automorphic orbits in finitely generated groups |
| topic | Group Theory 20F69, 20F65, 20F28 |
| url | https://arxiv.org/abs/2604.18104 |