Smooth (non)rigidity of cusp-decomposable manifolds

TL;DR

This paper introduces cusp-decomposable manifolds, establishes their smooth rigidity, and characterizes their outer automorphism groups, showing they can be realized by diffeomorphisms.

Contribution

It defines cusp-decomposable manifolds and proves their smooth rigidity, also describing the structure of their outer automorphism groups.

Findings

Outer automorphism group is an extension of an abelian group by a finite group.

Elements of the abelian group are induced by diffeomorphisms similar to Dehn twists.

Outer automorphism group can be realized by diffeomorphisms.

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 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.