Nilpotent groups of automorphisms of families of Riemann surfaces (original) (raw)

NILPOTENT AUTOMORPHISM GROUPS OF RIEMANN SURFACES

Transactions of AMS, 1985

The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automor-phisms of a surface of genus g > 2 cannot exceed 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).

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...

BOUNDS FOR THE ORDER OF SUPERSOLUBLE AUTOMORPHISM GROUPS OF RIEMANN SURFACES

Proceedings of AMS, 1990

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 is any finite supersoluble group acting as the automorphism group of some compact Riemann surface Í2 of genus g > 2 , then: (i) If g = 2 then \G\ < 24 and equality occurs when G is the supersoluble group At ® Z3 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 T has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24. (ii) If g > 3 then \G\ < 18(#-1), and if |G| = lS(g-1) then (g-1) must be a power of 3. Conversely if (g-1) = 3" , n > 2 , then there is at least one surface fí of genus g with an automorphism group of order 18(^-1) which must be supersoluble since its order is of the form 2-3m. 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.

On branched covering of compact Riemann surfaces with automorphisms

1997

In this work, we give an algor'ithm to count the different conforma/ equivalence classes of compact Riemann surfaces that admit a group of automorphisms isomorphic to Z/nZ, n E N, and that are branched coverings ofthe Riemann sphere, with signature ((n,O);m 1 ,m 2 ,m 3), m 1 ,mz,m3 E N. By using the previous result, we count the different conforma/ equivalence classes of compact Riemann surfaces in the cases of coverings with signature ((p,O);p,p,p), p 2:: 5 and prime, and signature ((p2,0);p 2 ,p 2 ,p), p 2::3 and prime.

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

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.

Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Annales Fennici Mathematici

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder,Širáň and Tucker for maps.

Analytic computation of some automorphism groups of Riemann surfaces

Kodai Mathematical Journal, 2007

Equations for the locus of Riemann Surfaces of genus three with a nonabelian automorphism group generated by involutions are determined from vanishings of Riemann's theta function. Torelli's Theorem implies that all of the properties of a non-hyperelliptic compact Riemann Surface (complex algebraic curve) X are determined by its period matrix W. This paper shows how to compute the group Aut X of conformal automorphisms of a surface X of genus three using W, in the case when the group is nonabelian and generated by its involutions. The connection between W and X is Riemann's theta function yðz; WÞ. Accola ([1], [2], [3]), building on classical results about hyperelliptic surfaces, found relationships between the theta divisor Y ¼ fz A JacðX Þ : yðz; WÞ ¼ 0g and Aut X. In the case of genus three, certain vanishings of y at quarter-periods of JacðX Þ imply that X has an automorphism s of degree two (or involution) such that X =hsi has genus one (making s an elliptic-hyperelliptic involution). This work derives equations in the moduli space of surfaces of genus three for many of the loci consisting of surfaces with a given automorphism group. It is a two-step process. First, topological arguments determine the order of the dihedral group generated by two non-commuting involutions. Then, combinatorial arguments about larger groups generated by involutions determine the theta vanishings corresponding to each. Much of the work here is based on the author's 1981 PhD dissertation [7] at Brown University. It appears now because of renewed interest in these questions, some of which is inspired by questions in coding theory: See [3], [5]. The research was directed by R. D. M. Accola, and Joe Harris was also a valuable resource. The author extends his (belated) thanks to them. 1. Preliminaries and notation In all that follows, X is a compact Riemann Surface (or complex algebraic curve) of genus three with automorphism group Aut X , period matrix W, jacobian 394