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

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.