Rigidifying simplicial complexes and realizing group actions

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.