Finite group actions on handlebodies and equivariant Heegaard genus for 3-manifolds

Abstract

Every finite group of symmetries (homeomorphisms) of a compact bounded surface of algebraic genus g acts, by taking the product with the interval [0,1], also on the 3-dimensional handlebody V g of genus g. In both cases, the maximal possible order of such a group is 12( g−1), and we call such a group a maximal bounded surface respectively handlebody group. Here we construct the first examples of maximal handlebody groups which are not maximal bounded surface groups. Our examples lead in a natural way to the notion of an equivariant Heegaard genus for finite group actions on 3-manifolds; we compute this genus for some classes of interesting examples. The notion of an equivariant Heegaard genus gives a certain hierarchy for finite group actions on a 3-manifold and the notion of a maximally symmetric 3-manifold; we show that the irreducible maximally symmetric 3-manifolds belong to the hyperbolic geometry, or are Seifert fibered of a very special type.