Context-Free Groups and Bass-Serre Theory

TL;DR

This paper surveys various characterizations of context-free groups, emphasizing their connections and providing a self-contained presentation of the Muller-Schupp theorem linking these groups to formal language theory.

Contribution

It offers a comprehensive survey of characterizations of context-free groups and presents a self-contained proof of the Muller-Schupp theorem without relying on Stallings' theorem.

Findings

Virtually free groups are exactly the context-free groups.

Multiple characterizations of context-free groups are interconnected.

A self-contained proof of the Muller-Schupp theorem is provided.

Abstract

The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In a seminal paper Muller and Schupp showed the converse: A context-free group is virtually free. Over the past decades a wide range of other characterizations of context-free groups have been found. The present notes survey most of these characterizations. Our aim is to show how the different characterizations of context-free groups are interconnected. Moreover, we present a self-contained access to the Muller-Schupp theorem without using Stallings' structure theorem or a separate accessibility result. We also give an introduction to some classical results linking groups with formal language theory.