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

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.