On automorphisms and splittings of special groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910066706219008 |
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| author | Fioravanti, Elia |
| author_facet | Fioravanti, Elia |
| contents | We initiate the study of outer automorphism groups of special groups $G$, in the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an infinite-order "generalised Dehn twist". Similarly, the coarse-median preserving subgroup ${\rm Out}_{\rm cmp}(G)$ is infinite if and only if $G$ splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist.
The proof is based on constructing and analysing non-small, stable $G$-actions on $\mathbb{R}$-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of $G$.
As a result of independent interest, we determine when generalised Dehn twists associated to splittings of $G$ preserve the coarse median structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13212 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On automorphisms and splittings of special groups Fioravanti, Elia Geometric Topology Group Theory We initiate the study of outer automorphism groups of special groups $G$, in the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an infinite-order "generalised Dehn twist". Similarly, the coarse-median preserving subgroup ${\rm Out}_{\rm cmp}(G)$ is infinite if and only if $G$ splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist. The proof is based on constructing and analysing non-small, stable $G$-actions on $\mathbb{R}$-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of $G$. As a result of independent interest, we determine when generalised Dehn twists associated to splittings of $G$ preserve the coarse median structure. |
| title | On automorphisms and splittings of special groups |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2108.13212 |