A family of two generator non-Hopfian groups

TL;DR

This paper constructs an infinite family of 2-generator non-Hopfian groups with specific presentations satisfying small cancellation conditions, expanding the understanding of non-Hopfian groups in geometric group theory.

Contribution

It introduces a new family of non-Hopfian groups with explicit presentations linked to 2-bridge link groups, demonstrating their properties under small cancellation conditions.

Findings

Constructed groups are non-Hopfian for all m ≥ 3.

Groups satisfy small cancellation conditions C(4) and T(4).

Presentations relate to 2-bridge link groups with specific continued fractions.

Abstract

We construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \dots$, where each $G_m$ has a specific presentation $G_m=\langle a, b \, | \, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \cdots =1 \rangle$ 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 $r_{m,i}=[m+1,m-1,(i-1)\langle m \rangle,m+1,m]$ in continued fraction expansion for every integer $i \ge 1$.