The central tree property and algorithmic problems on subgroups of free groups
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909312110034944 |
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| author | Roy, Mallika Ventura, Enric Weil, Pascal |
| author_facet | Roy, Mallika Ventura, Enric Weil, Pascal |
| contents | We study the average case complexity of the uniform membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14697 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The central tree property and algorithmic problems on subgroups of free groups Roy, Mallika Ventura, Enric Weil, Pascal Group Theory Computational Complexity 20E05, 20F10, 68Q17 We study the average case complexity of the uniform membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group. |
| title | The central tree property and algorithmic problems on subgroups of free groups |
| topic | Group Theory Computational Complexity 20E05, 20F10, 68Q17 |
| url | https://arxiv.org/abs/2303.14697 |