Solving the isomorphism problems for two families of parafree groups

Abstract

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $⟨x,y,z\mid x=[z^m,x][z^n,y]⟩$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $⟨x,y,z\mid x=[x^m,z^n][y,z]⟩$. 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 $m^{\prime }=m,n^{\prime }=n$.

18 pages

References in corpus (3)

• Topology and geometry of cohomology jump loci
• Bieri-Neumann-Strebel-Renz invariants and homology jumping loci
• Which 3-manifold groups are Kähler groups?