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 handlcbody of genus g which have genus zero quotient. Let q(z) = #SIGMA#_(g >= 0) Q_gz~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.
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机译：对于有限群G和非负整数g，令Q_g表示具有零商属的g属种的手体上保持方向的G动作的q等价类的数量。令q（z）= #SIGMA #_（g> = 0）Q_gz〜g是关联的生成函数。当G最多有一次对合时，我们证明q（z）是有理函数，其极点是单位根。我们证明了部分相反的情况，表明当G有多个对合时，q（z）是不合理的，或者在开放盘中有极点{| z | <1}。在G最多有一次对合的情况下，我们通过分析体现有关生成G的多集信息的有限姿态来获得Q_g的渐近近似。当G为循环时，可以找到更好的近似。
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