Generating Functions for Actions on Handlebodies with Genus Zero Quotient

For a finite group G and a nonnegative integer g, let Qg 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 Qgzg 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 Qg by analyzing a finite poset which embodies information about generating multisets of G. A finer approximation is found when G is cyclic.