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

中文翻译：

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求解两族超自由群的同构问题

对于任何整数 $米,n$ 和 $米\ne 0$ 和 $n>0$， 让 $G_{米,n}$ 表示由 $〈X,是,z|X=[z^{米},X][z^n,是]〉$; 对于任何整数$米,n>0$， 让 $H_{米,n}$ 表示由 $〈X,是,z|X=[X^{米},z^n][是,z]〉$. 通过研究不可约的上同调跳跃轨迹$\mathrm{GL}(2,\mathbb{C})$-字符品种，我们显示：如果 $米,{米}^{\prime }\ne 0$, $n,n^{\prime }>0$ 和 $G_{{米}^{\prime },n^{\prime }}\cong G_{米,n}$， 然后 $米={米}^{\prime },n=n^{\prime }$; 如果$米,{米}^{\prime },n,n^{\prime }>0$ 和 $H_{{米}^{\prime },n^{\prime }}\cong H_{米,n}$， 然后 ${米}^{\prime }=米,n^{\prime }=n$.