Symmetries of exotic aspherical space forms

Abstract

We study finite group actions on smooth manifolds of the form $M\#\Sigma$, where $\Sigma$ is an exotic $n$-sphere and $M$ is a closed aspherical space form. We give a classification result for free actions of finite groups on $M\#\Sigma$ when $M$ is 7-dimensional. We show that if $\mathbb Z/p\mathbb Z$ acts freely on $T^n\#\Sigma$, then $\Sigma$ is divisible by $p$ in the group of homotopy spheres. When $M$ is hyperbolic, we give examples $M\#\Sigma$ that admit no nontrivial smooth action of a finite group, even though Isom($M$) is arbitrarily large. Our proofs combine geometric and topological rigidity results with smoothing theory and computations with the Atiyah--Hirzebruch spectral sequence.