Some one-relator Hopfian groups

Abstract:

The group presented by \[ (a,t;{t^{ - 1}}{a^l}t = {a^m})\] is non-Hopfian if $l,m \ne \pm 1$ and $\pi (l) \ne \pi (m)$, where $\pi (l)$ and $\pi (m)$ denote the sets of prime divisors of l and m. By contrast, we prove that if w is a word of the free group $F({a_1},{a_2})$ which is not primitive and not a proper power, then the group $({a_1},{a_2},t;{t^{ - 1}}{w^l}t = {w^m})$ is Hopfian.