An algorithm to recognize H 3-manifold groups
Organizers
Speaker
Time
Tuesday, May 27, 2025 1:00 PM - 2:00 PM
Venue
A3-4-301
Online
Zoom 482 240 1589
(BIMSA)
Abstract
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 $< x_1,\dots,x_n ; r_1,\dots,r_m>$, where $\pi_1(H) = < x_1,\dots, x_n>$, 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 $G = < x_1,\dots, x_n ; r_1,\dots, r_m>$, 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 $G = < x_1,\dots, x_n ; r_1,\dots, r_m>$, 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.
Speaker Intro
Professor Fengchun Lei received his Ph.D. from Jilin University in 1990, once was a professor in Dalian University of Technology, and has joined in BIMSA from March, 2025. His research interests include 3-manifold topology, knot theory and topological data analysis. He ever won the second prize of Science and Technology Progress Award of the State Education Commission, and was selected into the "Cross-Century Excellent Talent Program" of the State Ministry of Education.