A remark on the fundamental group of a compact negatively curved manifold

Abstract

In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong to $\mathcal{C},$ where $\mathcal{C}$ is the smallest class of groups that contains all amenable groups and is closed under free products and finite extensions.