Generators for automorphisms of special groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910006036660224 |
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| author | Fioravanti, Elia |
| author_facet | Fioravanti, Elia |
| contents | Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[μ])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22789 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generators for automorphisms of special groups Fioravanti, Elia Group Theory Geometric Topology Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[μ])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument. |
| title | Generators for automorphisms of special groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2601.22789 |