Gluing locally symmetric manifolds: asphericity and rigidity

TL;DR

This paper introduces a method to construct aspherical manifolds called piecewise locally symmetric manifolds, analyzes their properties, and proves their homotopy equivalences are homotopic to homeomorphisms, revealing complex group structures.

Contribution

It develops a new construction technique for aspherical manifolds using reflection groups and explores their homotopy and group properties, expanding understanding of non-positively curved spaces.

Findings

Constructed new examples of aspherical manifolds

Proved homotopy equivalences are homotopic to homeomorphisms

Identified complex group structures in their fundamental groups

Abstract

We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fundamental groups have the structure of a complex of groups. These manifolds typically do not admit a locally $\CAT(0)$ metric. We prove that any self homotopy equivalence of such manifolds is homotopic to a homeomorphism. 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.