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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.07374 |
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| _version_ | 1866909134705655808 |
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| author | Hamann, Matthias |
| author_facet | Hamann, Matthias |
| contents | In this paper, we prove that infinite cancellative finitely generated hyperbolic monoids never contain $\mathbb N\times\mathbb N$ as a submonoid but that they contain an element of infinite order and, if they are elementary, then they also contain a free monoid of rank at least 2. As a corollary we obtain that the latter have exponential growth. We prove these results by analysing the monoid of self-embeddings of hyperbolic digraphs and proving fixed-point theorems for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free submonoids of hyperbolic monoids Hamann, Matthias Group Theory Combinatorics In this paper, we prove that infinite cancellative finitely generated hyperbolic monoids never contain $\mathbb N\times\mathbb N$ as a submonoid but that they contain an element of infinite order and, if they are elementary, then they also contain a free monoid of rank at least 2. As a corollary we obtain that the latter have exponential growth. We prove these results by analysing the monoid of self-embeddings of hyperbolic digraphs and proving fixed-point theorems for them. |
| title | Free submonoids of hyperbolic monoids |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2403.07374 |