Theta Constants, Riemann Surfaces and the Modular Group

2001 ◽  
Author(s):  
Hershel Farkas ◽  
Irwin Kra
Keyword(s):  
Riemann Surfaces ◽  
Modular Group

A modular group and Riemann surfaces of genus 2

1975 ◽  
Vol 142 (3) ◽  
pp. 205-219 ◽  
Author(s):  
Linda Keen
Keyword(s):  
Riemann Surfaces ◽  
Modular Group ◽  
Genus 2

scholarly journals Perfect Non-Extremal Riemann Surfaces

2000 ◽  
Vol 43 (1) ◽  
pp. 115-125 ◽  
Author(s):  
Paul Schmutz Schaller
Keyword(s):  
Riemann Surface ◽  
Quadratic Forms ◽  
Riemann Surfaces ◽  
Closed Geodesic ◽  
Modular Group ◽  
Infinite Family ◽  
Positive Definite ◽  
Normal Subgroups

AbstractAn infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group PSL(2, ℤ) of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface M the length of a systole, which is a shortest closed geodesic of M.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

On mappings with finite length distortion on Riemann surfaces

2019 ◽  
Vol 33 ◽  
Author(s):  
Serhii Volkov ◽  
Vladimir Ryazanov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Main Tool ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Lipschitz Mappings ◽  
Geometric Function ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

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