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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.07868 |
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| _version_ | 1866909241635241984 |
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| author | Bagnoud, Pierre Alderic Bodart, Corentin |
| author_facet | Bagnoud, Pierre Alderic Bodart, Corentin |
| contents | The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07868 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dead ends and rationality of complete growth series Bagnoud, Pierre Alderic Bodart, Corentin Group Theory The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either. |
| title | Dead ends and rationality of complete growth series |
| topic | Group Theory |
| url | https://arxiv.org/abs/2210.07868 |