Examples of aspherical manifolds, the reflection group trick and Borel-Serre compactifications

Aspherical manifolds are manifolds that have contractible universal covers, or equivalently, those that have trivial higher homotopy groups. Examples include nonpositively curved manifolds (by Cartan-Hadamard), and in particular, the rich class of locally symmetric spaces of non compact type. However, it is in general not easy to construct aspherical manifolds. e.g. the obvious connect sum operation usually ruins asphericity. One way to construct aspherical manifolds is to use the "reflection group trick". I will describe how to construct examples of aspherical manifolds by applying the reflection group trick to the Borel-Serre compactifications of locally symmetric spaces. This will be an example-oriented talk and everything will be explained via examples.