Centralizers of torsion in 3-manifold groups

Abstract

We show that if $P$ is a $P{D}_3$-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 $P\cong R{P}^2\times S^1$.

v2: added three questions relating to possible analogues of the Projective Plane Theorem, and a new section giving alternative arguments due (separately) to J.S. Crisp and G.A. Swarup