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

TL;DR

This paper classifies finite group actions on certain 3-manifolds with genus two Heegaard splittings and non-trivial JSJ-decompositions, showing they are mostly limited to cyclic or dihedral groups unless specific disk component conditions occur.

Contribution

It provides a classification of handlebody-preserving finite group actions on genus two Haken manifolds with non-trivial JSJ-decompositions, identifying when the symmetry group is restricted to  or 2.

Findings

Finite group actions are isomorphic to  or 2 under specified conditions.

The classification depends on the intersection properties of JSJ-tori with handlebodies.

Exceptions occur when certain intersection components are three or more disks.

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 smooth actions 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 $\mathbb{D}_2$ unless $V_j\cap(\cup T_i)$ has three or more disk components for $j=1$ or 2.