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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.04932 |
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| _version_ | 1866912147160694784 |
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| author | Bellingeri, Paolo Godelle, Eddy Paris, Luis |
| author_facet | Bellingeri, Paolo Godelle, Eddy Paris, Luis |
| contents | A new family of groups, called trickle groups, is presented. These groups generalize right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a presentation with relations of the form $xy = zx$ and $x^μ= 1$, that are governed by a simplicial graph, called a trickle graph, endowed with a partial ordering on the vertices, a vertex labeling, and an automorphism of the star of each vertex. We show several examples of trickle groups, including extended cactus groups, certain finite-index subgroups of virtual cactus groups, Thompson group F, and ordered quandle groups. A terminating and confluent rewriting system is established for trickle groups, enabling the definition of normal forms and a solution to the word problem. An alternative solution to the word problem is also presented, offering a simpler formulation akin to Tits' approach for Coxeter groups and Green's for graph products of cyclic groups. A natural notion of a parabolic subgraph of a trickle graph is introduced. The subgroup generated by the vertices of such a subgraph is called a standard parabolic subgroup and it is shown to be the trickle group associated with the subgraph itself. The intersection of two standard parabolic subgroups is also proven to be a standard parabolic subgroup. If only relations of the form $xy = zx$ are retained in the definition of a trickle group, then the resulting group is called a preGarside trickle group. Such a group is proved to be a preGarside group, a torsion-free group, and a Garside group if and only if its associated trickle graph is finite and complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trickle groups Bellingeri, Paolo Godelle, Eddy Paris, Luis Group Theory A new family of groups, called trickle groups, is presented. These groups generalize right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a presentation with relations of the form $xy = zx$ and $x^μ= 1$, that are governed by a simplicial graph, called a trickle graph, endowed with a partial ordering on the vertices, a vertex labeling, and an automorphism of the star of each vertex. We show several examples of trickle groups, including extended cactus groups, certain finite-index subgroups of virtual cactus groups, Thompson group F, and ordered quandle groups. A terminating and confluent rewriting system is established for trickle groups, enabling the definition of normal forms and a solution to the word problem. An alternative solution to the word problem is also presented, offering a simpler formulation akin to Tits' approach for Coxeter groups and Green's for graph products of cyclic groups. A natural notion of a parabolic subgraph of a trickle graph is introduced. The subgroup generated by the vertices of such a subgraph is called a standard parabolic subgroup and it is shown to be the trickle group associated with the subgraph itself. The intersection of two standard parabolic subgroups is also proven to be a standard parabolic subgroup. If only relations of the form $xy = zx$ are retained in the definition of a trickle group, then the resulting group is called a preGarside trickle group. Such a group is proved to be a preGarside group, a torsion-free group, and a Garside group if and only if its associated trickle graph is finite and complete. |
| title | Trickle groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2412.04932 |