Symmetries of exotic aspherical space forms

TL;DR

This paper investigates finite group actions on exotic aspherical manifolds, providing classification results, examples of manifolds with no nontrivial symmetries, and combining geometric, topological, and spectral sequence methods.

Contribution

It offers new classification results for free finite group actions on exotic aspherical manifolds, especially in 7 dimensions, and constructs examples with no nontrivial symmetries.

Findings

Finite group actions on $M ext{ extperiodcentered}\Sigma$ are classified in certain cases.

If $ ext{Z}/p ext{Z}$ acts freely on $T^n ext{ extperiodcentered}\Sigma$, then $ ext{ extSigma}$ is divisible by $p$.

Examples of hyperbolic manifolds with no nontrivial finite group actions despite large isometry groups.

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.