A family of two generator non-Hopfian groups

Abstract

We construct $2$-generator non-Hopfian groups $G_m,m=3,4,5,\dots$, where each $G_m$ has a specific presentation $G_m=⟨a,b|u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}=\cdots =1⟩$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)⟨m⟩,m+1,m]$ in continued fraction expansion for every integer $i\ge 1$.

18 pages, 1 figure