Extending finite group actions from surfaces to handlebodies

We show that every action of a finite dihedral group on a closed orientable surface $\mathcal F$ extends to a 3-dimensional handlebody $\mathcal V$, with $\partial \mathcal V=\mathcal F$. In the case of a finite abelian group $G$, we give necessary and sufficient conditions for a $G$-action on a surface to extend to a compact $3$-manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order $84(g-1)$ of $PSL(2,27)$ on a surface $\mathcal F$ of genus $g=118$ does not extend to any compact 3-manifold $M$ with $\partial M=\mathcal F$, thus resolving the only case of Hurwitz actions of type $PSL(2,q)$ of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137--151).