Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913816164433920 |
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| author | Bridson, Martin R. |
| author_facet | Bridson, Martin R. |
| contents | We prove a variety of fixed-point theorems for groups acting on CAT$(0)$ spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of $Γ$ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension ${\rm{FixDim}}(Γ)+1$ in which various groups of geometric interest can act on a complete CAT$(0)$ space without a global fixed point. For automorphism groups of free groups, we prove ${\rm{FixDim}}({\rm{Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups Bridson, Martin R. Group Theory Geometric Topology Metric Geometry 20F65, 20F67, 57M60, 20F28 We prove a variety of fixed-point theorems for groups acting on CAT$(0)$ spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of $Γ$ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension ${\rm{FixDim}}(Γ)+1$ in which various groups of geometric interest can act on a complete CAT$(0)$ space without a global fixed point. For automorphism groups of free groups, we prove ${\rm{FixDim}}({\rm{Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$. |
| title | Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups |
| topic | Group Theory Geometric Topology Metric Geometry 20F65, 20F67, 57M60, 20F28 |
| url | https://arxiv.org/abs/2505.00943 |