Parafree graphs of groups with cyclic edge groups

TL;DR

This paper provides a criterion for when the fundamental group of a finite graph of groups with cyclic edge groups is parafree, advancing understanding of their structure and relation to profinite rigidity.

Contribution

It establishes a new combination theorem characterizing parafree groups arising from graphs of groups with cyclic edges.

Findings

Identifies conditions for fundamental groups to be parafree

Connects parafree groups to profinite rigidity conjectures

Advances structural understanding of parafree groups

Abstract

We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with Remeslennikov's conjecture on the profinite rigidity of free groups. In this article, we determine when the fundamental group of a finite graph of groups with cyclic edge groups is parafree.