Automorphism Groups Of Riemann Surfaces Research Articles

Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface X_C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface X_C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface X_C is called symmetric and the curve C is real.

The occurrence of finite simple permutation groups of fixity 2 as automorphism groups of Riemann surfaces

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we study finite simple groups from a permutation action point of view. We classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most two fixed points on each non-regular orbit and at most four fixed points in total.

On discrete versions of two Accola’s theorems about automorphism groups of Riemann surfaces

In this paper we give a few discrete versions of Robert Accola’s results on Riemann surfaces with automorphism groups admitting partitions. As a consequence, we establish a condition for \(\gamma \)-hyperelliptic involution on a graph to be unique. Also we construct an infinite family of graphs with more than one \(\gamma \)-hyperelliptic involution.

Chiral covers of hypermaps

Generalising a conjecture of Singerman, it is shown that there are orientably regular chiral hypermaps (equivalently regular chiral dessins d'enfants) of every non-spherical type. The proof uses the representation theory of automorphism groups of Riemann surfaces acting on homology and on various spaces of differentials. Some examples are given.

On gonality automorphisms of p-hyperelliptic Riemann surfaces

A compact Riemann surface X of genus g > 1 is saID to be a p-hyperelliptic if X admits a conformal involution ρ for which X/ρ has genus p. This notion is the particular case of so called cyclic (q, n)-gonal surface which is defined as the one admitting a conformal automorphism δ of order n such that X/δ has genus q. It is known that for g > 4 p + 1, ρ is unique And so central in the automorphism group of X. We give necessary And sufficient conditions on p And g for the existence of a Riemann surface of genus g admitting commuting p-hyperelliptic involution ρ And (q, n)-gonal automorphism δ for some prime n And we study its group of automorphisms And the number of fixed points of δ. Furthermore, we deal with automorphism groups of Riemann surfaces admitting central automorphism with at most 8 fixed points. The condition on the small number of fixed points of such an automorphism is justified by the study of p-hyperelliptic surfaces.

On gonality automorphisms of p-hyperelliptic Riemann surfaces

A compact Riemann surface X of genus g > 1 is saID to be a p-hyperelliptic if X admits a conformal involution ρ for which X/ρ has genus p. This notion is the particular case of so called cyclic (q, n)-gonal surface which is defined as the one admitting a conformal automorphism δ of order n such that X/δ has genus q. It is known that for g > 4 p + 1, ρ is unique And so central in the automorphism group of X. We give necessary And sufficient conditions on p And g for the existence of a Riemann surface of genus g admitting commuting p-hyperelliptic involution ρ And (q, n)-gonal automorphism δ for some prime n And we study its group of automorphisms And the number of fixed points of δ. Furthermore, we deal with automorphism groups of Riemann surfaces admitting central automorphism with at most 8 fixed points. The condition on the small number of fixed points of such an automorphism is justified by the study of p-hyperelliptic surfaces.

A Multivariate Arithmetic Function of Combinatorial and Topological Significance

AbstractWe investigate properties of a multivariate function

Analytic computation of some automorphism groups of Riemann surfaces

Analytic computation of some automorphism groups of Riemann surfaces

On a theorem of supersoluble automorphism groups

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc. Amer. Math. Soc. 108 (1990), 587-600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that.

Realizing countable groups as automorphism groups of Riemann surfaces

Every countable group can be realized as the full automorphism group of a Riemann surface as well as the full group of isometries of a Riemannian manifold.

Symmetries of Accola-Maclachlan and Kulkarni surfaces

For all g ≥ 2 g \ge 2 there is a Riemann surface of genus g g whose automorphism group has order 8 g + 8 8g+8 , establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large g g the Accola-Maclachlan surface was unique for g = 0 , 1 , 2 mod 4 g= 0,1,2\mod 4 and produced exactly one additional surface (the Kulkarni surface) for g = 3 mod 4 g= 3\mod 4 . In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

Bounds for the Order of Supersoluble Automorphism Groups of Riemann Surfaces

Bounds for the Order of Supersoluble Automorphism Groups of Riemann Surfaces

Bounds for the order of supersoluble automorphism groups of Riemann surfaces

The maximal automorphism groups of compact Riemann surfaces for a class of groups positioned between nilpotent and soluble groups is investigated. It is proved that if G G is any finite supersoluble group acting as the automorphism group of some compact Riemann surface Ω \Omega of genus g ≥ 2 g \geq 2 , then: (i) If g = 2 g = 2 then | G | ≤ 24 |G| \leq 24 and equality occurs when G G is the supersoluble group D 4 ⊗ Z 3 {D_4} \otimes {{\mathbf {Z}}_3} that is the semidirect product of the dihedral group of order 8 and the cyclic group of order 3. This exceptional case occurs when the Fuchsian group Γ \Gamma has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24. (ii) If g ≥ 3 g \geq 3 then | G | ≤ 18 ( g − 1 ) |G| \leq 18\left ( {g - 1} \right ) , and if | G | = 18 ( g − 1 ) |G| = 18\left ( {g - 1} \right ) then ( g − 1 ) \left ( {g - 1} \right ) must be a power of 3. Conversely if ( g − 1 ) = 3 n , n ≥ 2 \left ( {g - 1} \right ) = {3^n},n \geq 2 , then there is at least one surface Ω \Omega of genus g g with an automorphism group of order 18 ( g − 1 ) 18\left ( {g - 1} \right ) which must be supersoluble since its order is of the form 2 3 m 2{3^m} . This bound corresponds to a specific Fuchsian group given by the signature (0;2,3,18). The terms in the chief series of each of these Fuchsian groups to the point where a torsion-free subgroup is reached are computed.

Nilpotent automorphism groups of Riemann surfaces

The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automorphisms of a surface of genus g ⩾ 2 g \geqslant 2 cannot exceed 16 ( g − 1 ) 16(g - 1) . Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0;2,4,8).