Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866908907361796096 |
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| author | Fioravanti, Elia |
| author_facet | Fioravanti, Elia |
| contents | We show that, under weak assumptions, the automorphism group of a ${\rm CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_08493 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes Fioravanti, Elia Geometric Topology Group Theory We show that, under weak assumptions, the automorphism group of a ${\rm CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex. |
| title | Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2001.08493 |