On the complex analytic theory of riemann surfaces and planar discontinuous groups

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.

Download Full-text

A Primer on Mapping Class Groups (PMS-49)

10.23943/princeton/9780691147949.001.0001 ◽

2017 ◽

Cited By ~ 1

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.

Download Full-text

On the family of Riemann surfaces with tetrahedral group action

Kodai Mathematical Journal ◽

10.2996/kmj/1572487229 ◽

2019 ◽

Vol 42 (3) ◽

pp. 476-495

Author(s):

Ryota Hirakawa

Keyword(s):

Group Action ◽

Riemann Surfaces ◽

The Family

Download Full-text

On mappings with finite length distortion on Riemann surfaces

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2019-33-4 ◽

2019 ◽

Vol 33 ◽

Cited By ~ 1

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.

Download Full-text

On mappings with finite length distortion on Riemann surfaces

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-1-4 ◽

2020 ◽

Vol 17 (1) ◽

pp. 58-59

Author(s):

Igor Protasov

Keyword(s):

Finite Length ◽

Riemann Surfaces ◽

Length Distortion

Download Full-text

The Volume of the Quiver Vortex Moduli Space

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptab012 ◽

2021 ◽

Author(s):

Kazutoshi Ohta ◽

Norisuke Sakai

Keyword(s):

Gauge Theory ◽

Moduli Space ◽

Riemann Surfaces ◽

Gauge Theories ◽

Target Space ◽

Contour Integral ◽

Quiver Gauge Theory ◽

Volume Formula ◽

Volume Operator ◽

Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

Download Full-text