Symmetries of exotic smoothings of aspherical space forms

Mauricio Bustamante, Bena Tshishiku

Published 2021-09-19Version 1

We study finite group actions on smooth manifolds of the form $W\#\Sigma$, where $\Sigma$ is an exotic $n$-sphere and $W$ is either a hyperbolic or a flat manifold. In the flat case, we classify the finite cyclic groups that act freely on an exotic torus $T^n\#\Sigma$. For hyperbolic manifolds $M$, we produce examples $M\#\Sigma$ that admit no nontrivial smooth action of a finite group, while Isom($M$) is arbitrarily large.