Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime (original) (raw)

AUTOMORPHISM GROUPS OF RIEMANN SURFACES OF GENUS p+1, WHERE p IS PRIME

Glasgow Mathematical Journal, 2005

We show that if S is a compact Riemann surface of genus g = p + 1, where p is prime, with a group of automorphisms G such that |G| ≥ λ(g − 1) for some real number λ > 6, then for all sufficiently large p (depending on λ), S and G lie in one of six infinite sequences of examples. In particular, if λ = 8 then this holds for all p ≥ 17.

On Riemann surfaces of genus g with 4g–4 automorphisms

Israel Journal of Mathematics

In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maximal order; namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g − 4. Under the assumption that g − 1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

Riemann surfaces and restrictively-marked hypermaps

If S is a compact Riemann surface of genus g > 1, then S has at most 84(g − 1) (orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of automorphisms of S and |G| > 24(g − 1) then G is the automorphism group of a regular oriented map (of genus g) and if |G| > 12(g − 1) then G is the automorphism group of a regular oriented hypermap of genus g (Singerman). We generalise these results and prove that if |G| > g − 1 then G is the automorphism group of a regular restrictedly-marked hypermap of genus g. As a special case we also show that a marked finite transitive permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.

Automorphisms of the Quot Schemes Associated to Compact Riemann Surfaces

International Mathematics Research Notices, 2013

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of O ⊕r X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on C r. The group PGL(r, C) acts on Q via the action of GL(r, C) on O ⊕r X. We prove that this subgroup PGL(r, C) is the connected component, containing the identity element, of the holomorphic automorphism group Aut(Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.

Riemann surface with cyclic automorphisms group

Proyecciones (Antofagasta), 1997

In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .

Symmetries of Riemann surfaces with large automorphism group

Mathematische Annalen, 1974

A Riemann surface is symmetric if it admits an anti-conformal involution. The basic question which we discuss in this paper is whether compact Riemann surfaces of genus g > t which admit large groups of automorphisms are symmetric. As is weU-known, the automorphism group of a compact Riemann surface of genus g > 1 is finite and bounded above by 84(g-1). Macbeath ([t21 13]) has found infinitely many g for which this bound is attained. We show that all the surfaces found by Macbeath's methods are indeed symmetric. However, we do exhibit an example of a non-symmetric Riemann surface of genus g = ! 7 which does admit 84(g-1) automorphisms. We also study Riemann surfaces admitting automorphisms of large order. The order of an automorphism of a Riemann surface of genus g is bounded above by 4g + 2 and this bound is attained for every g [8]. We show that all Riemann surfaces admitting automorphisms of order greater that 2g + 2 are symmetric. There is a close link between our work and the theory of irreflexible regular maps on surfaces. (See § 8 for definitions.) There is a connection between the groups of regular maps and large groups of automorphisms of compact Riemann surfaces. Indeed, every group of automorphisms ofa Riemann surface of genus g of order greater than 24(g-1) is also the group of some regular map and conversely, every group of a regular map can be thought of as the group of automorphisms of a Riemann surface. The irreflexible regular maps turn out to be rather exceptional. (In fact, it was suggested in early editions of [3] that they did not exist for surfaces of genus O > 1). We show in the above correspondence that large groups of automorphisms of non-symmetric surfaces will give rise to irreflexible regular maps, but that the converse of this fact is not always true. Thus, for example, groups of automorphisms of order greater than 24(g-1) of a compact non-symmetric Riemann surface of genus g are more exceptional than irreflexible regular maps. There is another interpretation of symmetric Riemann surfaces which is of interest. Every compact Riemann surface can be obtained as the Riemann surface of an algebraic curve f(z, w) = 0. A Riemann surface

Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

Glasgow Mathematical Journal, 1987

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence abo...

On automorphisms groups of cyclic p-gonal Riemann surfaces

Journal of Symbolic Computation, 2013

In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p ≥ 3 is a prime integer and the genus of the surfaces is at least (p − 1) 2 + 1. We use Fuchsian and NEC groups, and cohomology of finite groups.

On the one-dimensional family of Riemann surfaces of genus q with 4q automorphisms

Journal of Pure and Applied Algebra

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ 2 different from 3, 6, 12, 15 and 30, with exactly 4g automorphisms form an equisymmetric one-dimensional family, denoted by Fg. In this paper, for every prime number q ≥ 5, we explore further properties of each Riemann surface S in Fq as well as of its Jacobian variety JS.