Classification of generalized torsion elements of order two in 3-manifold groups

TL;DR

This paper classifies 3-manifolds whose fundamental groups contain generalized torsion elements of order two, and explores properties of R-groups within these groups, providing a comprehensive understanding of their algebraic structure.

Contribution

It introduces a classification of 3-manifolds with fundamental groups having order-two generalized torsion elements and establishes the equivalence of R-groups and R-groups in this context.

Findings

Classified 3-manifolds with order-two generalized torsion elements.

Proved R-group and R-group equivalence for 3-manifold groups.

Identified which 3-manifold groups are R-groups.

Abstract

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is called the order of $g$. We will classify $3$-manifolds $M$, each of whose fundamental group has a generalized torsion element of order two. Furthermore, we will classify such elements in $\pi_1(M)$. We also prove that $R$-group and $\overline{R}$-group coincide for $3$-manifold groups, and classify $3$-manifold groups which are $R$-groups (and hence $\overline{R}$-groups).