# Characters and automorphism groups of compact Riemann surfaces

• 199 Pages
• 0.57 MB
• English
by
Cambridge University Press , Cambridge, U.K, New York
Riemann surfaces, Automorphisms, Characters of g
Classifications The Physical Object Statement Thomas Breuer Series London Mathematical Society lecture note series -- 280 LC Classifications QA333 .B74 2000 Pagination xii, 199 p. : Open Library OL16987691M ISBN 10 0521789094, 0521798094 LC Control Number 00058590

This book deals with automorphism groups of compact Riemann surfaces, of genus at least two, viewed as factor groups of Fuchsian groups. The author uses modern methods from computational group theory and representation theory, providing classifications of all automorphism groups up to genus The book also classifies the ordinary characters for several groups, arising from the action of automorphisms on the space of holomorphic abelian differentials of a compact Reimann by: Automorphisms of compact Riemann surfaces Generation of groups Classification for small genus Classification for fixed group: real characters Classification for fixed group: nonreal irrationalities.

Series Title: London Mathematical Society lecture note. Get this from a library. Characters and automorphism groups of compact Riemann surfaces.

[Thomas Breuer]. Characters and Automorphism Groups of Compact Riemann Surfaces by Thomas Breuer Lehrstuhl D fur Mathematik RWTH, Aachen, Germany E-mail: [email protected] Errata p. 6, l. Write \The map" not \So the map".

8, l. Insert a comma before c r. 20, l. -4 to Theorems 3A and 3B in [Gre63] are not correct, so replace. 3. Breuer, T.: Characters and Automorphism Groups of Compact Riemann Surfaces.

LMS Lecture Notes, vol. Cambridge University Press, Cambridge ()Cited by: 2. As far as I know, one of the best references for your question is the book of Breuer. Characters and Automorphism Groups of Compact Riemann Surfaces. It contains calculations for $2 \leq g \leq 48$.

Many of them have been obtained by using the Computer Algebra System GAP. Automorphism groups of compact Riemann surfaces paper, all groups are solvable except when n = (for g= 3) and n = 60, (for g = 4) and so we can apply results on normalizers for the construction of groups. Note that the maximum of orders of abelian groups of automorphisms of Riemann.

This monograph deals with symmetries of compact Riemann surfaces. A symmetry of a compact Riemann surface S is an antianalytic involution of S. It is well known that Riemann surfaces.

Hurwitz states that the maximal automorphism group of a compact Riemann surface of genus 9 has order at most 84(g-1). It is well-known that the Klein quartic is the unique genus 3 curve that attains the Hurwitz Author: Nima Anvari.

JOURNAL OF ALGEBRA() Automorphism Groups of Compact Riemann Surfaces of Genus Five AKIKAZU KURIBAYASHI Department of Mathematics, Chuo University, TokyoJapan AND HIDEYUKI KIMURA Department of Mathematics, Tokyo Institute of Characters and automorphism groups of compact Riemann surfaces book, TokyoJapan Communicated by Walter Feit Received August 8, Let X be a compact Riemann surface of Cited by: On compact Riemann surfaces with dihedral groups of automorphisms Article (PDF Available) in Mathematical Proceedings of the Cambridge Philosophical Society (03) May with Reads.

Characters and Automorphism Groups of Compact Riemann Surfaces. Part of London Mathematical Society Lecture Note Series. AUTHOR: Thomas Breuer. DATE PUBLISHED: October FORMAT: Paperback.

ISBN: While there appears to. ] AUTOMORPHISM GROUPS ON COMPACT RIEMANN SURFACES 3. New lower bounds. Our method will be to realize certain abstract groups as a group of cover transformations of a surface covering the Riemann sphere.

Let IF0 be the Riemann sphere and let E0 = {qx, a2, a3} be a set of three points on it. Let Wq=W0 — E0. BibTeX @MISC{Breuer_errataet, author = {Thomas Breuer}, title = {Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces}, year = {}}.

2 Paula Tretkoﬁ Among the most interesting compact Riemann surfaces are those with a group G of automorphisms which is relatively large compared with general problem of determining all such surfaces S and groups G is very di–cult, but it tends to be easier when the Euler characteristic ´ = 2(1¡g) of S has a simple numerical form.

A second paper of Belolipetsky and. Let X be a compact Riemann surface of genus g>1 and let G be a group of biholomorphic mappings on X onto itself. Consider all pairs (X,G).We say that (X,G) is topologically equivalent to (X′,G′) if there exists an o.p.

(orientation preserving) homeomorphism h of X onto X′ such that G′h= this paper, we shall classify the (X,G)'s up to topological equivalence in the case g= by: [Book:Topics in the Theorem of Riemann surfaces] Let S be a compact Riemann surface of genus g.

Suposse S admit a nite group of automorphism G 0 and G 0 = S s i=1 G i where G i is a subgroup of G 0 for each i.

Let the order of G i be p i, let S i = S=G iand let n i. One prefers to consider compact Riemann surfaces and thus the compact-iﬁcation Cˆ is called the Riemann surface of the curve C. It turns out that all compact Riemann surfaces can be described as com-pactiﬁcations of algebraic curves (see for example [Jos06]).

Quotients under group actions Deﬁnition 4. Let ∆be a domain in C. range 1 compact Riemann surfaces of genus g with this maximum possible number of automorphisms.

The automorphism group of each such surface is isomorphic to a quotient A/N of the (2, 3,7) triangle group A = (x, y 1 x2 = y3 = (xy)’ = 1), where N is one of 92 proper normal subgroups of A with index less than In the second case you get the punctured unit disc.

(Thus these are the only hyperbolic riemann surfaces whose automorphism group is of positive dimension) Hence any compact hyperbolic surface X has discrete automorphism group, which then has to be finite because of compactness of X.

(Any orbit in X is discrete, hence finite since X is compact). This has been used several times to compute automorphism groups for compact Riemann surfaces (for example see [2] or [13]).

By using those methods, it can be proved that T S is a normal subgroup. Characters and Automorphism Groups of Compact Riemann Surfaces: Background information to the book with this title. Multiplicity-Free Permutation Characters: An online version of the classification of multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups (joint work with Klaus Lux).

The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry: genus 0 – the isometry group of the sphere is the Möbius group of projective transforms of the complex line.

For many more details, see Breuer’s book “Characters and automorphism groups of compact Riemann surfaces”.

### Details Characters and automorphism groups of compact Riemann surfaces EPUB

Given a compact Riemann surface (curve) X and ﬁnite group G =Aut(X), let Y = X=G be the set of orbits of X under the action of G.

The genus of Y. Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero).One of the first facts one observes in their theory is that the group of automorphisms of is quite large when has genus zero or one.

When has genus zero, it is the projective line, and its automorphism group is, a fact which generalizes naturally to higher projective spaces. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The question of whether a given group G which acts faithfully on a compact Riemann surface X of genus g 2 is the full group of automorphisms of X (or some other such surface of the same genus) is considered.

Conditions are derived for the extendability of the action of the group G in terms of a concrete partial. Bounds for the Order of Symmetry Group of Automorphism of Compact Riemann Surface automorphism groups",Indian J. Pure Appl. Math 12(11), pp () - Chutia C.

### Description Characters and automorphism groups of compact Riemann surfaces EPUB

Solutions to some problems on Riemann surface automorphism groups. Thesis () - Chetiya B. and Chutia C., Genera of Compact Riemann surfaces with Dihedral. Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus We decompose the Jacobian varieties of hyperelliptic curves up to ge deﬁned over an algebraically closed ﬁeld of characteristic zero, with reduced automorphism group A 4, S 4, or A 5.

Among these curves is a genus-4curve. EQUATIONS OF RIEMANN SURFACES WITH AUTOMORPHISMS 3 In the sequel we will be primarily interested in large automorphism groups, that is, jAut(X)j>4(g X 1). In this case, the Riemann-Hurwitz formula implies that g 0 = 0 and 3 r 4.

Surface kernel generators are used in the following formula for the number of xed points of an automor-phism. The representations of the automorphism group of a compact Riemann surface on the first homology group and the spaces of (/-differentials are decom-posed into irreducibles.

As an application it is shown that A/24 is not a Hurwitz group. Introduction. Let G be a finite group of orientation-preserving homeomor-phisms of a Riemann surface S of. In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1).

A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface.

The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1].Cited by: We say a Riemann surface has a large automorphism group if #Aut(C) > 4(g − 1).

We find equations of all genus 4 and 5 Riemann surfaces C with large automorphism groups, and all but two of the genus 6 Riemann surfaces with large automorphisms groups. The strategy used is based on the Eichler trace formula and the projection formula from the character theory of finite groups.