Smooth (non)rigidity of cusp-decomposable manifolds

T. Tam Nguyen Phan

Published 2009-07-22, updated 2011-10-18Version 3

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 a manifold is an extension of an abelian group by a finite group. Elements of the abelian group are induced by diffeomorphisms that are analogous to Dehn twists in surface topology. We also prove that the outer automophism group can be realized by a group of diffeomorphisms of the manifold.