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Topological classification of conformal actions on 2-hyperelliptic Riemann surfaces. (English) Zbl 1087.30039

The classification of conformal actions on compact Riemann surfaces up to topological conjugacy is a classical and difficult problem, which has been solved just in very few particular cases. For surfaces of genus \(g=2,3\) this question was treated in [S. A. Broughton, “Classifying finite groups actions on surfaces of low genus”, J. Pure and Appl. Algebra 69, No. 3, 233–270 (1991; Zbl 0722.57005)], and for genus \(g=4\) in [O. V. Bogopolski, “Classifying the actions of finite groups on orientable surfaces of genus 4”, Sib. Adv. Math. 7, No. 4, 9–38 (1997; Zbl 0926.57011)]. Moreover, the actions on elliptic-hyperelliptic surfaces were classified by the author of the present article in [E. Tyszkowska, “Topological classification of conformal actions on elliptic-hyperelliptic Riemann surfaces”, J. Algebra 288, 345–363 (2005; Zbl 1078.30037)].
The paper under review is devoted to solve the same problem for the class of 2-hyperelliptic Riemann surfaces, i.e. surfaces \(X\) of genus \(g>2\) whose quotient under a conformal involution \(\rho\) is an orbifold of genus two. To be precise, the author restricts herself to surfaces of genus \(g> 9\) since under this assumption the involution \(\rho\) is unique and it commutes with all automorphisms of \(X.\)
The arguments in the article, which is very well written and organized, combine some standard strategies from the combinatorial theory of Riemann surfaces (e.g., the Riemann-Hurwitz formula, the uniformization by means of Fuchsian groups), but used in a clever way, with very intricate group theoretic tricks requiring a big amount of skill.
It is worthwhile mentioning that the author also determines exactly the maximal actions, that is, the cases in which the group acting on \(X\) is its full automorphism group. To that end she uses the celebrated list of nonmaximal signatures due to [D. Singerman, “Finitely generated maximal Fuchsian groups”, J. Lond. Math. Soc. 6, II. Ser, 29–38 (1972; Zbl 0251.20052)] and some “ad hoc” arguments to get a set of generators of a Fuchsian group \(\Lambda\) in terms of a canonical set of generators of a Fuchsian group \(\Gamma\) containing \(\Lambda\) as a subgroup of index 2.
It is also remarkable the use of GAP Programme to compute from suitable presentations the order of some finite groups.

MSC:

30F20 Classification theory of Riemann surfaces
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