Generating Functions for Actions on Handlebodies with Genus Zero Quotient

Abstract

For a finite group G and a nonnegative integer g, let Q g denote the number of q-equivalence classes of orientation-preserving G-actions on the handlebody of genus g which have genus zero quotient. Let q( z)=∑ g⩾0 Q g z g be the associated generating function. When G has at most one involution, we show that q( z) is a rational function whose poles are roots of unity. We prove a partial converse showing that when G has more than one involution, q( z) is either irrational or has a pole in the open disk {| z|<1}. In the case where G has at most one involution, we obtain an asymptotic approximation for Q g by analyzing a finite poset which embodies information about generating multisets of G. A finer approximation is found when G is cyclic.