Rigidifying simplicial complexes and realizing group actions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912598400696320 |
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| author | Costoya, Cristina Gomes, Rafael Viruel, Antonio |
| author_facet | Costoya, Cristina Gomes, Rafael Viruel, Antonio |
| contents | We show that any action of a finite group on a finitely presentable group arises as the action of the group of self-homotopy equivalences of a space on its fundamental group. In doing so, we prove that any finite connected (abstract) simplicial complex $\mathbf{K}$ can be rigidified -- meaning it can be perturbed in a way that reduces the full automorphism group to any subgroup -- while preserving the homotopy type of the geometric realization $| \mathbf{K} |$. We also obtain that every action of a finite group on a finitely generated abelian group is the action of the group of self-homotopy equivalences of a space on one of its higher homotopy groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidifying simplicial complexes and realizing group actions Costoya, Cristina Gomes, Rafael Viruel, Antonio Algebraic Topology Combinatorics Group Theory 55P10 (Primary) 55U99, 20B25, 06A06, 06A11 (Secondary) We show that any action of a finite group on a finitely presentable group arises as the action of the group of self-homotopy equivalences of a space on its fundamental group. In doing so, we prove that any finite connected (abstract) simplicial complex $\mathbf{K}$ can be rigidified -- meaning it can be perturbed in a way that reduces the full automorphism group to any subgroup -- while preserving the homotopy type of the geometric realization $| \mathbf{K} |$. We also obtain that every action of a finite group on a finitely generated abelian group is the action of the group of self-homotopy equivalences of a space on one of its higher homotopy groups. |
| title | Rigidifying simplicial complexes and realizing group actions |
| topic | Algebraic Topology Combinatorics Group Theory 55P10 (Primary) 55U99, 20B25, 06A06, 06A11 (Secondary) |
| url | https://arxiv.org/abs/2509.09646 |