Last edited by Grogor
Saturday, August 1, 2020 | History

5 edition of Symmetric and Alternating Groups As Monodromy Groups of Riemann Surfaces 1 found in the catalog.

# Symmetric and Alternating Groups As Monodromy Groups of Riemann Surfaces 1

## by Robert M. Guralnick

Written in English

Subjects:
• Algebraic geometry,
• Groups & group theory,
• Mathematics,
• Curves,
• Monodromy groups,
• Permutation groups,
• Science/Mathematics

• Edition Notes

The Physical Object ID Numbers Contributions R. Stafford (Contributor) Format Paperback Number of Pages 128 Open Library OL11420219M ISBN 10 0821839926 ISBN 10 9780821839928

Lecture 1 What are Riemann surfaces? Problem: Natural algebraic expressions have ‘ambiguities’ in their solutions; that is, they de ne multi-valued rather than single-valued functions. In the real case, there is usually an obvious way to x this ambiguity, by selecting one branch of the function. For example, consider f(x) = p x. Chapter Some applications farther a eld from nite groups Polynomial subgroup-growth in nitely-generated groups Relative Brauer groups of eld extensions Monodromy groups of coverings of Riemann surfaces Some exotic applications more brie y treated Locally nite simple groups and Moufang loops

Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points with an Appendix by R. Guralnick and R. Staﬀord Robert M. Guralnick, University of Southern California, Los Angeles, CA, and John . $\begingroup$ Dear Mark: I am sure there are many excellent graduate students at Chicago in algebra, algebraic geometry, algebraic topology, analysis, differential geometry, logic, number theory, PDEs, probability, and representation theory who do not know what a quasi-isometry is. Moreover, you must know this is the case, since you wouldn't use quasi-isometries in a colloquium talk without.

See Theorem of "Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points", AMS Memoirs (), vol. , no. This explains why you were able to find the examples you found.   An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, .

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### Symmetric and Alternating Groups As Monodromy Groups of Riemann Surfaces 1 by Robert M. Guralnick Download PDF EPUB FB2

Get this from a library. Symmetric and alternating groups as monodromy groups of Riemann surfaces 1: Generic covers and covers with many branch points. [Robert M Guralnick; John Shareshian].

The authors consider indecomposable degree $$n$$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $$d$$. They show that if the cover has five or more branch points then the genus grows rapidly with $$n$$ unless either $$d = n$$ or the curves have genus zero, there are precisely five branch points and.

Symmetric and alternating groups as monodromy groups of Riemann surfaces I: generic covers and covers with many branch points. I would proceed following the description you give in the following way, first the branch points are $0,-1$ and $\infty$, so if we shade the points that are sent to the upper plane under the map $$z \to \frac{4z^2(z-1)^2}{(2z-1)^2}$$ we obtain a triangulation of the sphere were the ramification points will be placed in the vertex and the.

The fact that symmetric groups of all orders can appear as monodromy groups of Riemann surfaces of genus zero is a long-standing one. In this paper, a further search has been made in order to determine which linite groups G can and cannot appear as monodromy groups of Riemann surfaces of genus zero.

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 Cited by: with Riemann surfaces.

Riemann surfaces are central in mathematics because of the multiple connections between complex analysis, algebraic geometry, hyperbolic geometry, group theory, topology etc. The main focus is the connection of holomorphic morphisms with branched coverings, and the use of permutation groups in classifying these Size: KB.

Fuchsian groups, and we also prove an analogue of Higman’s conjecture for symmetric quotients, determining precisely which Fuchsian groups surject to all but ﬁnitely many symmetric groups (Theorem ). These results have applications to monodromy groups of.

Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space. An example of a non-Riemannian symmetric space is anti-de Sitter space.

Algebraic definition. Let G be a connected Lie group. Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks Article in Journal of Algebra (2) June with 25 Reads How we measure 'reads'.

The classical fact concerning symmetric surfaces is () Harnack's Theorem. IF J is an anticonformal involution on the closed Riemann surface S of genus g, then the fixed point set of J is either empty or consists of 5+1 disjoint simple curves cQ,cs where 0.

In this context the pertinence of the results in [5], in this paper and in [6] relies on the fact that the most common Galois groups are the symmetric and the alternating groups [15].

Finally, we. Abstract. Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§), we expect there are only finitely many monodromy groups of primitive genus 0 by: 3.

10 CHAPTER 1. HOLOMORPHIC FUNCTIONS The second integral is deﬁned for all z, and holomorphic in z. We write the ﬁrst integral as Z1 0 tz−1(et−1)dt+ Z1 0 tz−1dt.

Now the term Z 1. Discrete Groups and Riemann Surfaces Anthony Weaver July 6, Abstract These notes summarize four expository lectures delivered at the Ad-vanced School of the ICTS Program Groups, Geometry and Dynamics, De-cember,Almora, India.

The target audience was a group of students at or near the end of a traditional undergraduate math major. My File Size: KB. JOURNAL OF ALGEBRA() Maximal Automorphism Groups of Symmetric Riemann Surfaces with Small Genus MARSTON CONDER Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand Communicated by Peter M.

Neumann Received May 3, The genus of a finite group G is the smallest genus of its Cayley by: AroundKlein and Poincaré studied monodromy groups for linear differential equations on compact Riemann surfaces.

See Klein [4] and Poincaré [9, 10]. Roughly stated, there were two fundamental problems: (A) given the DE, to calculate the monodromy group; (B) to determine all DE [of a certain type] having a given monodromy group.

The complex plane C is the most basic Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for map g(z) = z * (the conjugate map) also defines a chart on C and {g} is an atlas for charts f and g are not compatible, so this endows C with two distinct Riemann surface structures.

In fact, given a Riemann surface X and its atlas A, the. Goals of the Lecture: To get an idea of the classification of Riemann surfaces that can be arrived at based on the fundamental group, using the theory of covering spaces - To get introduced to the notions of: moduli problem, moduli space, number of moduli, fine and coarse classification, and to write these down for simple Riemann surfaces Topics: Biholomorphic map or isomorphism of Riemann.

Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, MayThe papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmüller spaces, Jacobian varieties, and quasiconformal mappings.

These topics are intertwined, representing a common meeting of algebra. History of Riemann surfaces Daniel Ying ∗ Octo Figure 1: Bernhard Riemann, () Abstract Riemann surfaces have an appealing feature to mathematicians (and hope-fully to non-mathematicians as well) in that they appear in a variety of mathe-matical ﬁelds.

The point of the introduction of Riemann surfaces made by Riemann, KleinFile Size: KB.Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question.

Provide details and share your research! But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations.1 Riemann surfaces Deﬁnitions and examples Let Xbe a topological space.

We want it to look locally like C. So we make the following deﬁnition. Deﬁnition 1. A complex chart on X is a homeomorphism φ: U → V where Uis an open set in Xand V is an open set in File Size: KB.