Solving the isomorphism problem for two families of parafree groups
报告题目(Title)：Solving the isomorphism problem for two families of parafree groups
报告人(Speaker)：陈海苗 （北京工商大学）
地点(Place)：后主楼1220
时间(Time)：2023年3月31日 15:00-16:00
邀请人(Inviter)：程志云
报告摘要
Let $U$ be a linear group. For each group $\Gamma$, let $\mathcal{X}_{U}(\Gamma)$ denote the space of conjugacy classes of irreducible representations $\Gamma\to U$. Then $\mathcal{X}_{U}:\Gamma\mapsto \mathcal{X}_{U}(\Gamma)$ defines a contra-variant functor from the category of groups to that of topological spaces. In particular, $U={\rm GL}(2,\mathbb{C})$ and $U=\mathbb{C}^\times$ give rise to two functors. The homomorphism $\det:{\rm GL}(2,\mathbb{C})\to\mathbb{C}^\times$ induces a natural transformation $\det_\ast:\mathcal{X}_{{\rm GL}(2,\mathbb{C})}\to\mathcal{X}_{\mathbb{C}^\times}$. This means: if there exists an isomorphism $\phi:\Gamma'\to\Gamma$, then it induces homeomorphisms $\phi^\ast:\mathcal{X}_{{\rm GL}(2,\mathbb{C})}(\Gamma)\to\mathcal{X}_{{\rm GL}(2,\mathbb{C})}(\Gamma')$ and $\phi^\ast:\mathcal{X}_{\mathbb{C}^\times}(\Gamma)\to\mathcal{X}_{\mathbb{C}^\times}(\Gamma')$, which are consistent with $\det_\ast$.
In 1960's and 1990's, Baumslag introduced two families of parafree groups. For nonzero integers $m,n$, put $G_{m,n}=\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$, and $H_{m,n}=\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. About whether two members in $G$ or $H$ family are isomorphic, few results have been obtained till 2020. Using the above machinery, we show that, except for trivial cases, if $m\ne m'$ or $n\ne n'$, then $G_{m,n}\not\cong G_{m',n'}$ and $H_{m,n}\not\cong H_{m',n'}$.
