The $K(π,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909327981281280 |
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| author | Goldman, Katherine |
| author_facet | Goldman, Katherine |
| contents | Let $A_Γ$ be an Artin group with defining graph $Γ$. We introduce the notion of $A_Γ$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_Γ$ being extra-large relative to two parabolic subgroups, one of which is always large type. Under this new condition, we show that $A_Γ$ satisfies the $K(π,1)$ conjecture whenever each of the distinguished subgroups do. In addition, we show that $A_Γ$ is acylindrically hyperbolic under only mild conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16391 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $K(π,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups Goldman, Katherine Group Theory Metric Geometry 20F36 (Primary) 20F65, 57M60 (Secondary) Let $A_Γ$ be an Artin group with defining graph $Γ$. We introduce the notion of $A_Γ$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_Γ$ being extra-large relative to two parabolic subgroups, one of which is always large type. Under this new condition, we show that $A_Γ$ satisfies the $K(π,1)$ conjecture whenever each of the distinguished subgroups do. In addition, we show that $A_Γ$ is acylindrically hyperbolic under only mild conditions. |
| title | The $K(π,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups |
| topic | Group Theory Metric Geometry 20F36 (Primary) 20F65, 57M60 (Secondary) |
| url | https://arxiv.org/abs/2211.16391 |