Grzegorz Gromadzki

Automorphism Groups of Compact Bordered Klein Surfaces

Preliminary results.- Klein surfaces as orbit spaces of NEC groups.- Normal NEC subgroups of NEC groups.- Cyclic groups of automorphisms of compact Klein surfaces.- Klein surfaces with groups of automorphisms in prescribed families.- The automorphism group of compact Klein surfaces with one boundary component.- The automorphism group of hyperelliptic compact Klein surfaces with boundary.

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g.

On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces

Abstract An old theorem of Harnack states that a symmetry of a compact Riemann surface X of genus g , ( g ≥ 2) has at most g + 1 disjoint simple closed curves of fixed points, each of which is called the oval of X . Much more recently Natanzon proved that for v ( g ) being the maximum number of ovals that a surface of genus g admits, v ( g ) ≤ 42( g − 1). We show in this paper that actually for g ≠ 2,3,5,7,9, v ( g ) ≤ 12( g − 1), that this bound is sharp for infinitely many g and we calculate v ( g ) for the mentioned above exceptional values of g as well.

Symmetries of Riemann surfaces on which PSL(2, q) acts as a Hurwitz automorphism group

Let X be a compact Riemann surface and Aut(X) be its automorphism group. An automorphism of order 2 reversing the orientation is called a symmetry. The authors together with D. Singerman have been working on symmetries of Riemann surfaces in the last decade. In this paper, the symmetry type St(X) of X is defined as an unordered list of species of conjugacy classes of symmetries of X, and for a class of particular surfaces, St(X) is found. This class consists of Riemann surfaces on which PSL(2, q) acts as a Hurwitz group. An algorithm to calculate the symmetry type of this class is provided.

On compact Riemann surfaces with dihedral groups of automorphisms

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group DN. The question of extendability of the action of DN is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

ON OVALS OF NON-CONJUGATE SYMMETRIES OF RIEMANN SURFACES

We find the upper bound for the total number of ovals of k (5 ≤ k ≤ 8) non-conjugate symmetries of a Riemann surface of genus g, without an additional assumption concerning their separabilities, filling this way the gap existing in known literature of the subject. We also prove that our bound is attained for infinitely many values of g. Finally, as a byproduct of our considerations, we obtain the algebraic structure of the group generated by the extremal configurations of the symmetries, provided it is a 2-group.

ON REAL FORMS OF A COMPLEX ALGEBRAIC CURVE

Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X (R) and Y (R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X , that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X . In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

Supersoluble groups of automorphisms of compact Riemann surfaces

Given an integer g ≥ 2 and a class of finite groups let N ( g , ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p -groups (given p ), soluble groups and finally for metabelian groups, an upper bound for N ( g , ) as well as infinite sequences for g for which this bound is attained were found in [ 5, 6, 7, 8, 13 ], [ 4 ], [ 10 ], [ 15 ], [ 16 ], [ 1 ], [ 2 ] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

SCHOTTKY UNIFORMIZATIONS OF SYMMETRIES

A real algebraic curve of genus g is a pair (S , 〈τ〉), where S is a closed Riemann surface of genus g and τ : S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S , 〈τ〉) is a tuple (Ω,Γ, P : Ω→ S ), where Γ is a Schottky group with region of discontinuity Ω and P : Ω→ S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Mobius transformation τ keeping Ω invariant with P ◦ τ = τ ◦ P. The extended Kleinian group K = 〈Γ, τ〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein-Maskit’s combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups we get alternative proofs to results due to Kalliongis and McCullough on orientation reversing involutions on handlebodies.

On Ovals of Riemann Surfaces of Even Genera

AbstractLet X be a Riemann surface of genus g≥2. A symmetry of σ of X is an antiholomorphic involution acting of X. A classical theorem of Harnack states that the set Fix (σ) of fixed points of σ is either emplty or it consists of ‖σ‖≤ g+1 disjoint simple closed curves called, following Hilbert′s terminology, the ovals of σ. A Riemann surface admitting a symmetry corresponds to a real algebraic curve and nonconjugate symmetries correspond to different real models of the curve. The number of ovals of the symmetry equals the number of connected components of the corresponding real model. It is well known that two symmetries of a Riemann surface of genus g have at most 2g+2 ovals, and the bound is attained for every genus and just for commuting symmetries. Natanzon showed that three and four nonconjugate symmetries of a Riemann surface of genus g have at most 2g+4 and 2g+8 ovals respectively, and these bounds are attained for every odd genus and for commuting symmetries. Natanzon found that a Riemann surface of genus g has at most 2(