Conjugacy Class Growth in Virtually Abelian Groups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929730386657280 |
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| author | Dermenjian, Aram Evetts, Alex |
| author_facet | Dermenjian, Aram Evetts, Alex |
| contents | We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06144 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conjugacy Class Growth in Virtually Abelian Groups Dermenjian, Aram Evetts, Alex Group Theory Combinatorics 05E16, 20E45, 20F55, 20F69 We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class. |
| title | Conjugacy Class Growth in Virtually Abelian Groups |
| topic | Group Theory Combinatorics 05E16, 20E45, 20F55, 20F69 |
| url | https://arxiv.org/abs/2309.06144 |