Solving the isomorphism problems for two families of parafree groups

TL;DR

This paper classifies isomorphisms among two families of parafree groups by analyzing their cohomology jump loci, establishing that the parameters defining these groups are uniquely determined by their isomorphism types.

Contribution

It provides a complete solution to the isomorphism problem for the groups $G_{m,n}$ and $H_{m,n}$ using cohomology jump loci analysis, a novel approach in this context.

Findings

Isomorphism of $G_{m,n}$ groups implies equal parameters $m,n$

Isomorphism of $H_{m,n}$ groups implies equal parameters $m,n$

Cohomology jump loci effectively distinguish these groups

Abstract

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$.