Extending Finite Group Actions from Surfaces to Handlebodies

We show that every action of a finite dihedral group on a closed orientable surface F extends to a 3-dimensional handlebody V , with ∂V = 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 F of genus g = 118 does not extend to any compact 3-manifold M with ∂M = 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).