Automorphisms of the Baumslag-Gersten group

TL;DR

This paper classifies automorphisms of the Baumslag-Gersten group, proving it is Hopfian and co-Hopfian, and describes its outer automorphism group, providing a self-contained and accessible exposition of known results.

Contribution

It offers a self-contained, accessible proof of the automorphism classification of the Baumslag-Gersten group, including its Hopfian and co-Hopfian properties.

Findings

The Baumslag-Gersten group is Hopfian.

It is co-Hopfian.

Outer automorphisms form a group isomorphic to the dyadic rationals.

Abstract

We classify homomorphisms of the Baumslag-Gersten group into itself. We prove it is Hopfian and co-Hopfian. We show that the group of outer automorphisms of the Baumslag-Gersten group is isomorphic to the dyadic rationals with the addition operation. These results are not new. They were obtained by Andrew M. Brunner in \cite{Brunner}. However, our exposition is self-contained and, hopefully, more accessible for some readers.