An algorithm to recognize H 3-manifold groups
数学学科创建110周年系列报告
报告题目(Title)：An algorithm to recognize H 3-manifold groups
报告人(Speaker)：雷逢春 （北京雁栖湖应用数学研究院）
地点(Place)：后主楼1124
时间(Time)：2025年6月27日 16:00-17:00
邀请人(Inviter)：高红铸
报告摘要
A finitely presented group $G$ is called a {\em 3-manifold group} if $G$ is isomorphic to the fundamental group of a compact connected 3-manifold. Stallings proved in 1960s that there is no algorithm to determine whether an arbitrary finitely presented group is a 3-manifold group. Let $H$ be a handlebody of genus $n\geq 1$, $\mathcal{J}=\{J_1,\cdots,J_m\}$ a collection of pairwise disjoint simple closed curves on $\partial H$. The manifold obtained by attaching 2-handles to $H$ along $\mathcal{J}$ and filling each of the resulting 2-sphere with a 3-ball (if any) is called a {\em $m$-relator} 3-manifold, and is denoted by $H_\mathcal{J}$. From the construction, $G=\pi_1(H_\mathcal{J})$ has a natural presentation as $\la x_1,\cdots,x_n; r_1,\cdots,r_m \ra$, where $\pi_1(H)=\la x_1,\cdots,x_n\ra$, and $r_i=[J_i]\in \pi_1(H)$ (after some conjugation) for $1\leq i \leq m$. Such a presentation is called an {\em H presentation} of $G$. If a group $G$ admits an H presentation, we call $G$ an {\em H 3-manifold group}. Here is our main result: For a given finitely presented group $G=\la x_1,\cdots,x_n; r_1,\cdots,r_m \ra$, there exists an algorithm to determine whether it is an H presentation (therefore an H 3-manifold group). In the talk, I will explain the idea to show the theorem. This is a joint work with Liyuan Ma, Liang Liang, Xuezhi Zhao, and Jie Wu.
