An algorithm to recognize H 3-manifold groups
演讲者
时间
2025年05月27日 13:00 至 14:00
地点
A3-4-301
线上
Zoom 482 240 1589
(BIMSA)
摘要
A finitely presented group $G$ is called a 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 $n \geq 1$, $\mathcal{J} = {J_1,\dots, J_m}$ 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 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 $\langle x_1,\dots,x_n ; r_1,\dots,r_m\rangle$, where $\pi_1(H) = \langle x_1,\dots, x_n\rangle$, and $r_i = [J_i] \in \pi_1(H)$ (after some conjugation) for $1 \leq i \leq m$. Such a presentation is called an H presentation of $G$. If a group $G$ admits an $H$ presentation, we call $G$ an $H$ 3-manifold group. Here is our main
result:
\textbf{Theorem} For a given finitely presented group $G = \langle x_1,\dots, x_n ; r_1,\dots, r_m\rangle$, 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.
result:
\textbf{Theorem} For a given finitely presented group $G = \langle x_1,\dots, x_n ; r_1,\dots, r_m\rangle$, 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.