On the Boone--Higman Conjecture for groups acting on locally finite trees

TL;DR

This paper introduces a new method to prove the Boone--Higman Conjecture for groups acting on locally finite trees, successfully applying it to Baumslag--Solitar and free-by-cyclic groups, and suggesting broader applicability.

Contribution

The authors develop a novel approach for verifying the Boone--Higman Conjecture in specific classes of groups acting on trees, expanding the conjecture's verified cases.

Findings

Proved the Boone--Higman Conjecture for all Baumslag--Solitar groups.

Proved the Boone--Higman Conjecture for all free(finite rank)-by-cyclic groups.

Demonstrated the method's potential for broader applications.

Abstract

We develop a method for proving the Boone--Higman Conjecture for groups acting on locally finite trees. As a consequence, we prove the Boone--Higman Conjecture for all Baumslag--Solitar groups and for all free(finite rank)-by-cyclic groups, solving it in two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky. We also illustrate that our method has applications beyond these cases and may offer a route for proving the Boone--Higman Conjecture for many classes of groups.