The stable genus increment for group actions on closed 2-manifolds
The paper under review deals with families of finite group actions on surfaces (compact, and usually connected, 2-manifolds without boundary) such that, for a finite group \(G\), the family of \(G\)-actions is closed under taking equivariant connected sums. The authors ask for the genera of surfaces admitting a \(G\)-action in the family, generalizing the Hurwitz problem of determining the genera of surfaces with an (effective) \(G\)-action [cf. \textit{A. Hurwitz}, ``Über algebraische Gebilde mit eindeutigen Transformationen in sich'', Math. Ann. 41, 403-442 (1892; JFM 24.0380.02)]. A number of authors determined the smallest genus of surfaces admitting \(G\)-actions for various classes of finite groups \(G\) [see, e.g., \textit{H. Glover} and \textit{D. Sjerve} [J. Reine Angew. Math. 380, 59- 86 (1987; Zbl 0613.57019)], and \textit{C. Maclachlan}, Proc. Lond. Math. Soc., III. Ser. 15, 699-712 (1965; Zbl 0156.089)]. The first broad result related with the Hurwitz problem is the stable solution of it due to \textit{R. S. Kulkarni} [Topology 26, 195-203 (1987; Zbl 0623.20029)], which, for a finite group \(G\), involves an integer (called here the stable genus increment for the family of group actions) depending on the type of the 2-Sylow subgroup of \(G\) and on the exponents and orders of the Sylow subgroups of \(G\). The authors develop a general theory allowing to determine and calculate (in many cases) stable genus increments for families of group actions on surfaces. In particular, under mild assumptions on the family, the stable genus increment equals the product of the increments of the Sylow subgroups of \(G\). To illustrate the methods and various stable genus increments results including calculations of bounds, the authors describe the complete solution of the Hurwitz problem for finite groups of prime order. In calculations of the bounds of increments, the authors make use of orbifolds, so that lower bounds follow from restrictions on orbifold Euler characteristics, and upper bounds are obtained by constructing actions at the level of their orbifold quotients.
Hurwitz problem for finite groups of prime order
scientific article; zbMATH DE number 89841