Generating functions for actions on handlebodies with genus zero quotient (Q1296757)

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 \(q(z)=\sum_{g \geq 0}Q_g z^g \) be the associated generating function. When \(G\) has at most one involution, the authors show that \(q(z)\) is a rational function whose poles are roots of unity. They 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, they obtain an asymptotic approximation for \(Q_g\) by analyzing a finite poset which embodies information about generating multisets of \(G\).