Handlebody-preserving finite group actions on Haken manifolds with Heegaard genus two

TL;DR

This paper classifies finite groups acting on certain 3-manifolds with genus two Heegaard splittings and non-trivial JSJ-decompositions, showing they are limited to specific small groups under certain intersection conditions.

Contribution

It provides a classification of handlebody-preserving finite group actions on genus two Haken manifolds with non-trivial JSJ-decompositions, identifying conditions under which the groups are isomorphic to Z2 or D2.

Findings

G is isomorphic to Z2 or D2 under specified conditions

Finite group actions are constrained by intersection properties of JSJ-tori

Classification applies to manifolds with genus two Heegaard splittings and non-trivial JSJ-decompositions

Abstract

Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for $i=1,2$. If $G$ is a finite group of orientation-preserving diffeomorphisms acting on $M$ which preserves each handlebody of the Heegaard splitting and each piece of the JSJ-decomposition of $M$, then $G\cong \mathbb{Z}_2$ or $\mathbb{D}_2$ if $V_j\cap(\cup T_i)$ consists of at most two disks or at most two annuli.