Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

TL;DR

This paper introduces piecewise rank 1 manifolds, a class of aspherical manifolds assembled from locally symmetric pieces, and establishes their smooth rigidity under certain conditions, while analyzing their self-homotopy equivalences.

Contribution

It defines piecewise rank 1 manifolds, proves their smooth rigidity when cusps' structures are preserved, and characterizes their self-homotopy equivalence groups.

Findings

Smooth rigidity holds when cusp structures are preserved.

Self-homotopy groups can contain infinite abelian subgroups.

Twists along cusp centers generate elements of the automorphism group.

Abstract

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements of this abelian subgroup are twists along elements in the center of the fundamental group of a cusp.