Rigidifying simplicial complexes and realizing group actions

Cristina Costoya; Rafael Gomes; Antonio Viruel

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arXiv · 2509.09646

Rigidifying simplicial complexes and realizing group actions

Cristina Costoya·Rafael Gomes·Antonio Viruel

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Abstract

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.

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Cristina Costoya, Rafael Gomes, Antonio Viruel. 2025-09-11. Rigidifying simplicial complexes and realizing group actions. https://arxiv.org/abs/2509.09646

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