Leonid Olegovich Chekhov

On quantizing Teichmüller and Thurston theories

In earlier work, Chekhov and Fock have given a quantization of Teichmuller space as a Poisson manifold, and the current paper first surveys this material adding further mathematical and other detail, including the underlying geometric work by Penner on classical Teichmuller theory. In particular, the earlier quantum ordering solution is found to essentially agree with an ``improved operator ordering given by serially traversing general edge-paths on a graph in the underlying surface. Now, insofar as Thurstons sphere of projectivized foliations of compact support provides a useful compactification for Teichmuller space in the classical case, it is natural to consider corresponding limits of appropriate operators to provide a framework for studying degenerations of quantum hyperbolic structures. After surveying the required background material on Thurston theory and ``train tracks, the current paper continues to give a quantization of Thurstons boundary in the special case of the once-punctured torus, where there are already substantial analytical and combinatorial challenges. Indeed, an operatorial version of continued fractions as well as the improved quantum ordering are required to prove existence of these limits. Since Thurstons boundary for the once-punctured torus is a topological circle, the main new result may be regarded as a quantization of this circle. There is a discussion of quantizing Thurstons boundary spheres for higher genus surfaces in closing remarks.We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincare dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Riemanns bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the complex of curves and the geometry of Teichmueller space.This survey article considers moduli of algebraic curves using techniques from the complex analytic Teichmuller theory of deformations for the underlying Riemann surfaces and combinatorial topology of surfaces. The aim is to provide a readable narrative, suitable for people with a little background in complex analysis, hyperbolic plane geometry and discrete groups, who wish to understand the interplay of combinatorial, geometric and topological processes in this area. We explore in some detail a natural relationship with Grothendieck dessins, which provides both an appropriate setting in which to describe Veech curves (a special type of Teichmuller disc) and also a framework for relating complex moduli to arithmetic data involving a field of definition for the associated algebraic curves.We study the boundary of Teichmueller disks in a partial compactification of Teichmueller space, and their image in Schottky space. We give a broad introduction to Teichmueller disks and explain the relation between Teichmueller curves and Veech groups. Furthermore, we describe Braungardts construction of this partial compactification and compare it with the Abikoff augmented Teichmueller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary. This prepares the description of boundary points that a Teichmueller disk has, with a particular emphasis to the case that it leads to a Teichmueller curve. Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmueller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmueller disks in Schottky space and show that one can choose the projection from Teichmueller space to Schottky space in such a manner that the image of the Teichmueller disk is a quotient by an infinite group.The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. nThe paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174