Centralizers of torsion in 3-manifold groups

TL;DR

This paper investigates the structure of 3-manifold groups with torsion elements, showing that such groups retract onto specific subgroups and characterizing the topology of certain 3-manifolds.

Contribution

It establishes a link between torsion elements with infinite centralizers in 3-manifold groups and their retraction properties, providing new insights into their algebraic and topological structure.

Findings

Groups with torsion and infinite centralizer retract onto Z/2Z ⊕ Z

Irreducible closed 3-manifolds with such properties are homeomorphic to RP^2 × S^1

Characterization of 3-manifold groups based on torsion and centralizer properties

Abstract

We show that if $P$ is a $PD_3$-complex and $g\in\pi_1(P)$ has finite order $>1$ and infinite centraliser then $\pi_1(P)$ retracts onto $Z/2Z\oplus\mathbb{Z}$. If $P$ is an irreducible closed 3-manifold then it follows from the Projective Plane Theorem that $P\cong{RP^2}\times{S^1}$.