On genus actions of finite simple groups on handlebodies and bounded surfaces

Let G be a finite nonabelian simple group of isometries of a 3-dimensional handlebody V or of a compact bounded surface of least possible genus g (a "genus action" of G). Generalizing analogous results for genus actions of finite simple groups on closed surfaces, we prove that G is a normal subgroup of small index in the orientation-preserving isometry group H of V, and that H is canonically isomorphic to a subgroup of the automorphism group of G. We present also some related results on finite group actions of maximal possible order, in particular for nilpotent and simple groups.