Smooth (non)rigidity of cusp-decomposable manifolds

Abstract

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite-volume, locally symmetric, negatively curved manifolds with cusps. We prove that the group of outer automorphisms of the fundamental group of such manifolds contains a free abelian normal subgroup whose elements are induced by diffeomorphisms that are analogous to Dehn twists in surface topology. For the case where the decomposition is finite, the group of outer automorphisms of the fundamental group is an extension of a finitely generated free abelian group by a finite group. We also prove that the outer automorphism group can be realized by a group of diffeomorphisms of the manifold.