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

TL;DR

This paper surveys results on the fundamental group of compact negatively curved manifolds, highlighting a theorem that classifies these groups as outside a specific class closed under certain group operations.

Contribution

It reviews a theorem showing that fundamental groups of such manifolds do not belong to a particular class of groups closed under free products and finite extensions.

Findings

Fundamental groups of compact negatively curved manifolds are outside class .

The class  is the smallest containing all amenable groups and closed under free products and finite extensions.

The paper discusses implications of Gusevskij's theorem for geometric group theory.

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.