Automorphisms Of Compact Riemann Surfaces Research Articles

Addendum to “On the orders of largest groups of automorphisms of compact Riemann surfaces” [J. Pure Appl. Algebra 225 (12) (2021) 106758

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . In the earlier paper we have showed together with Czesław Bagiński that there exist a rational sequence ( r n ) convergent to 8 and infinite sets R n so that μ ( g ) = r n ( g − 1 ) for g ∈ R n . Moreover, assuming the Twin Prime conjecture we have showed the same for 12 i.e. there exist a rational sequence ( s n ) convergent to 12 and infinite sets S n so that μ ( g ) = s n ( g − 1 ) for g ∈ S n . Here we prove the second assertion using Chen theorem instead of relying on TPC.

On the orders of largest groups of automorphisms of compact Riemann surfaces

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that μ ( g ) ranges between 8 ( g + 1 ) and 84 ( g − 1 ) and these two bounds are attained for infinitely many g . Here we prove that given a prime q > 167 congruent to −1 modulo 3, not congruent +1 modulo 5 and a prime p ≥ 84 q , μ ( g ) = ( 8 ( q + 4 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many integers m . Furthermore we also prove that for any prime q for which p = q + 2 is a prime, μ ( g ) = ( 12 ( q + 2 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many m . These mean that there exist an infinite rational sequence ( r n ) convergent to 8 and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence ( s n ) convergent to 12 together with infinite sets R n and S n of integers for which μ ( g ) = { r n ( g − 1 ) if g ∈ R n s n ( g − 1 ) if g ∈ S n and 8 , 12 are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of μ . Namely if instead of μ = μ ( g ) we consider its normalization μ ˜ ( g ) = μ ( g ) / g then { 8 } ⊆ ( M d ) d ⊆ { 8 , 12 } , where M stands for the set of values of μ ˜ and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, 12 ∈ ( M d ) d .

Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride

Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride

Bounds of the Group of Automorphisms of Compact Riemann Surface With Reference To The Point Group of Sulphur Molecule

Bounds of the Group of Automorphisms of Compact Riemann Surface With Reference To The Point Group of Sulphur Molecule

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch et al. (2013) [12] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an ‘equivalent’ form of Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.

Dihedral Groups of Order 2p of Automorphisms of Compact Riemann Surfaces of Genus p – 1

AbstractIn this paper we prove that there is only one conjugacy class of dihedral group of order 2p in the 2(p – 1) × 2(p – 1) integral symplectic group that can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus p – 1. A pair of representative generators of the realizable class is also given.

Dihedral groups of automorphisms of compact Riemann surfaces of genus two

In this short note, the conjugacy classes of finite dihedral subgroups of the 4 × 4 integral symplectic group are considered. A complete list of representatives of the classes is obtained, among them six classes are realizable by analytic automorphisms of compact connected Riemann surfaces of genus two.

Matrix Classes Realized by Automorphisms of Compact Riemann Surfaces of Genus Two

A complete list of realizable conjugacy classes of torsion in the 4 × 4 integral symplectic group by analytic automorphisms of compact connected Riemann surfaces of genus two is given.

Sets of periods for automorphisms of compact Riemann surfaces

Let G = 〈 f 〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g ≥ 2 . Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x ∈ S , x is fixed by f d but not by any smaller power of f . For an arbitrary subset A of proper divisors of N , there is always an associated action and, if g A denotes the minimal genus for such an action, an algorithm is obtained here to determine g A . Furthermore, a set A max is determined for which g A is maximal.

On the weights of the fixed points of an automorphism of a compact Riemann surface

In this work we study automorphisms of compact Riemann surfaces with more than four fixed points. We obtain a lower bound for the weight of each of these fixed points. The discussion depends on the parities of the order of the automorphism and the number of fixed points. Moreover, we discuss the sharpness of our bounds.

REGULAR-NILPOTENT GROUPS OF AUTOMORPHISMS

Using the definition of regular p-group given by M. Hall (1), a new class of finite groups called regular-nilpotent has been defined. The action of these groups as automorphisms of compact Riemann surfaces has been investigated. It is proved that an ecessary and sufficient condition for a Fuchsian group to cover a regular-nilpotent group is that its orbit genus be zero and its periods satisfy the least common multiple condition ,fi rst defined by Harvey (2 )a nd Maclachlan (4).

Characters and automorphism groups of compact Riemann surfaces

Preface Notation 1. Compact Riemann surfaces 2. Group characters 3. Automorphisms of compact Riemann surfaces 4. Generation of groups 5. Classification for small genus 6. Classification for fixed group: real characters 7. Classification for fixed group: nonreal irrationalities Appendix A. Abelian inariants Appendix B. Irreducible characters Appendix C. Maillet's determinant Bibliography Index.

Metacyclic groups of automorphisms of compact Riemann surfaces

Metacyclic groups of automorphisms of compact Riemann surfaces

Dihedral Groups of Automorphisms of Compact Riemann Surfaces

AbstractIn this note we determine which dihedral subgroups of GLg(ℂ) can be realized by group actions on Riemann surfaces of genusg> 1.

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:

Fixed points of automorphisms of compact Riemann surfaces and higher-order Weierstrass points

A sufficient condition for fixed points of an automorphism of prime order on a compact Riemann surface to be higher-order Weierstrass points is given. This leads us to a complete study of the cases where the prime orders are small. 1. Let M be a compact Riemann surface of genus g > 2. We denote Aut M the group of conformal automorphisms of M, v(T) the number of fixed points of an automorphism T E Aut M and Hq(M) the space of holomorphic q-differentials on M. Lewittes proved that if v(T) > 5, then every fixed point is a 1-Weierstrass point [5], and in this relation, some cases have been studied by Accola [1], Duma [2], Farkas and Kra [3] for higher-order Weierstrass points (see Corollaries 1, 2, 3, 4 below). Guerrero [4] proved that if v(T) = 1 and the fixed point is not a 1 -Weierstrass point, then T has order 6, g 1 mod 6 and the fixed point is a q-Weierstrass point for all q > 2. It is known that if the order of T is prime, then v(T) > 2 [3]. Guerrero also gave examples of Riemann surfaces with automorphisms of prime order whose two fixed points are not q-Weierstrass points for q > 2. The purpose of this paper is to give a sufficient condition for fixed points to be q-Weierstrass points (q > 2) and to supplement the results mentioned above. We will show that if v(T)(2s + 1 n) $4 2(n5 r), then the fixed points of T are q-Weierstrass points, and study the case where v (T) > 3 and the order of T is 5. 2. For T E Aut M, let e be the rotation constant of T at a fixed point of T, i.e. locally T : z -* ez. There is a basis for the space of holomorphic q-differentials such that the linear map induced by T on this space is given by the matrix diag(eYl +q 6 Y2-2+q ... ,6Yd l+q) for each q > 2, where d = (2q 1)(g 1), and 1 = Y1 < Y2 < ..< yd < 2q(g 1) + 2 is the q-gap Received by the editors June 27, 1988 and, in revised form, August 29, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14F07; Secondary 30F10. ( I1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

Fixed Points of Automorphisms of Compact Riemann Surfaces and Higher-Order Weierstrass Points

Fixed Points of Automorphisms of Compact Riemann Surfaces and Higher-Order Weierstrass Points

On Metabelian Groups of Automorphisms of Compact Riemann Surfaces

On Metabelian Groups of Automorphisms of Compact Riemann Surfaces

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).

A matrix representation for automorphisms of compact Riemann surfaces

In this paper we prove a result which has as corollaries theorems of Hurwitz, Accola, Grothendieck, and Serre on automorphisms of Riemann surfaces.