Groups of $\mathrm{I}_G$-type
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911020889407488 |
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| author | Dietzel, Carsten |
| author_facet | Dietzel, Carsten |
| contents | In this work, we address a question posed by Dehornoy et al. in the book "Foundations of Garside Theory" that asks for a theory of groups of $\mathrm{I}_G$-type when $G$ is a Garside group. In this article, we introduce a broader notion than the one suggested by Dehornoy et al.: given a left-ordered group $G$, we define a group of $\mathrm{I}_G$-type as a left-ordered group whose partial order is isomorphic to those of $G$. Furthermore, we develop methods to give a characterization of groups of $\mathrm{I}_Γ$-type in terms of skew braces when $Γ$ is an Artin-Tits group of spherical type and classify all groups of $\mathrm{I}_Γ$-type where $Γ$ is an irreducible spherical Artin-Tits group, therefore providing an answer to another question of Dehornoy et al. concerning $\mathrm{I}_{B_n}$ structures where $B_n$ is the braid group on $n$ strands with its canonical Garside structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Groups of $\mathrm{I}_G$-type Dietzel, Carsten Group Theory Rings and Algebras 06B99, 06F15, 20F36, 20N99 In this work, we address a question posed by Dehornoy et al. in the book "Foundations of Garside Theory" that asks for a theory of groups of $\mathrm{I}_G$-type when $G$ is a Garside group. In this article, we introduce a broader notion than the one suggested by Dehornoy et al.: given a left-ordered group $G$, we define a group of $\mathrm{I}_G$-type as a left-ordered group whose partial order is isomorphic to those of $G$. Furthermore, we develop methods to give a characterization of groups of $\mathrm{I}_Γ$-type in terms of skew braces when $Γ$ is an Artin-Tits group of spherical type and classify all groups of $\mathrm{I}_Γ$-type where $Γ$ is an irreducible spherical Artin-Tits group, therefore providing an answer to another question of Dehornoy et al. concerning $\mathrm{I}_{B_n}$ structures where $B_n$ is the braid group on $n$ strands with its canonical Garside structure. |
| title | Groups of $\mathrm{I}_G$-type |
| topic | Group Theory Rings and Algebras 06B99, 06F15, 20F36, 20N99 |
| url | https://arxiv.org/abs/2505.13347 |