Generating Functions for Actions on Handlebodies with Genus Zero Quotient
✍ Scribed by Matt Compton; Andy Miller
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 201 KB
- Volume
- 87
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
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.
1999 Academic Press
In the latter part of the nineteenth century, Felix Klein and Adolph Hurwitz made some interesting observations about the automorphism groups of closed Riemann surfaces. The results involve what we now refer to as the Hurwitz problem for a finite group G: to describe the set called the genus spectrum of G which consists of all integers g such that there is a Riemann surface of genus g which has G as its automorphism group. Today the problem can be rephrased as asking to determine those integers g for which the closed orientable smooth 2-manifold of genus g admits an effective action by G. The main observation of Hurwitz was that the number 1 84 |G| +1 is a lower bound for the genus spectrum of G. However the precise determination of the minimal element of the genus spectrum for a finite group G is a difficult problem which has been the focus of investigations of many authors since Hurwitz's time. Recent results have determined this minimal element for most of the finite simple groups. Beyond this, Kulkarni was the first to consider the much broader Hurwitz problem