2-dimensional Shephard groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916493125484544 |
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| author | Goldman, Katherine |
| author_facet | Goldman, Katherine |
| contents | The 2-dimensional Shephard groups are quotients of 2-dimensional Artin groups by powers of standard generators. We show that such a quotient is not $\mathrm{CAT}(0)$ if the powers taken are sufficiently large. However, for a given 2-dimensional Shephard group, we construct a $\mathrm{CAT}(0)$ piecewise Euclidean cell complex with a cocompact action (analogous to the Deligne complex for an Artin group) that allows us to determine other non-positive curvature properties. Namely, we show the 2-dimensional Shephard groups are acylindrically hyperbolic (which was known for 2-dimensional Artin groups), and relatively hyperbolic (which most Artin groups are known not to be). As an application, we show that a broad class of 2-dimensional Artin groups are residually finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15434 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 2-dimensional Shephard groups Goldman, Katherine Group Theory Algebraic Topology Metric Geometry 20F65 (Primary) 57M60, 20E26, 20F67 (Secondary) The 2-dimensional Shephard groups are quotients of 2-dimensional Artin groups by powers of standard generators. We show that such a quotient is not $\mathrm{CAT}(0)$ if the powers taken are sufficiently large. However, for a given 2-dimensional Shephard group, we construct a $\mathrm{CAT}(0)$ piecewise Euclidean cell complex with a cocompact action (analogous to the Deligne complex for an Artin group) that allows us to determine other non-positive curvature properties. Namely, we show the 2-dimensional Shephard groups are acylindrically hyperbolic (which was known for 2-dimensional Artin groups), and relatively hyperbolic (which most Artin groups are known not to be). As an application, we show that a broad class of 2-dimensional Artin groups are residually finite. |
| title | 2-dimensional Shephard groups |
| topic | Group Theory Algebraic Topology Metric Geometry 20F65 (Primary) 57M60, 20E26, 20F67 (Secondary) |
| url | https://arxiv.org/abs/2411.15434 |