Symmetries of exotic aspherical space forms

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We study finite group actions on smooth manifolds of the form M#Σ, where Σ 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#Σ when M is 7-dimensional. We show that if Z/pZ acts freely on T^n#Σ, then Σ is divisible by p in the group of exotic spheres. When M is hyperbolic, we give examples M#Σ 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.