Group Theoretical, Combinatorial, and Dynamical Aspects of Mapping Class Groups

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Abstract Award: DMS 1510556, Principal Investigator: Dan Margalit The main goal of this project is to study surfaces and their symmetries. A surface is a two-dimensional space, in other words, a two-dimensional version of the world we live in. Surfaces come in many shapes (for instance the surface of a ball is different from the surface of a doughnut) and they arise in many varied contexts, from physics to robotics to data analysis to quantum field theory. The symmetries of a surface form a beautiful and rich theory that has been the focus of intense study over the past century. One surprising phenomenon is that there are combinatorial objects called curve complexes - looking nothing themselves like a surface - that have the same symmetries as a surface. Many such objects have been discovered in the past twenty years. The first goal of this project is to give (essentially) a complete list of such curve complexes. This will be a capstone in the well-studied theory of symmetries of curve complexes. The second project is to study a certain centrally important subset of the set of symmetries of a surface - the so-called Torelli group. These symmetries are significant because of their strong connections to algebraic geometry and representation theory. Basic properties of the Torelli group are unknown, despite the fact that this group has been studied heavily for fifty years. This project aims to understand the basic finiteness properties of the Torelli group - for instance finite presentability. This is one of the main open problems in the theory of surfaces. Using a computer-aided search, new footholds have been found into this problem. The third project is a proposed algorithm for quickly computing the basic properties of a single symmetry of a surface. For instance, this algorithm computes the entropy, which is the amount of mixing being achieved on the surface. Other such algorithms exist, but ours is much faster. For instance, using an appropriate notion of size (called word length), the existing algorithms can handle symmetries of size 30 (or so) and our algorithm can very easily handle symmetries of size upwards of 30,000. In conjunction with these projects, the principal investigator will also be completing a textbook for undergraduates on a related subject, called Office Hours with a Geometric Group Theorist, and also will continue to run a professional development workshop for graduate students, the Topology Students Workshop. The mapping class group of a surface is the group of homotopy classes of orientation-preserving homeomorphisms of the surface. Among other things, the mapping class group encodes the outer automorphism group of the surface fundamental group, the (orbifold) fundamental group of the moduli space of the surface, and the isomorphism types of surface bundles over arbitrary spaces. The mapping class group also has connections to many, many areas of mathematics, including dynamics, group theory, number theory, quantum field theory, representation theory, and algebraic geometry, just to name a few. The goals laid out in this project are threefold: (1) find a general theory for when a combinatorial, algebraic, or geometric object associated to a surface has the extended mapping class group as its group of automorphisms; (2) determine the finiteness properties of the Torelli subgroup of the mapping class group, specifically whether or not the Torelli group is finitely presented; and (3) establish a polynomial-time algorithm to compute the conjugacy invariants for a pseudo-Anosov mapping class. Problem (1) was conceived by Ivanov; with Brendle, the PI has made substantial progress on this question. Problem (2) is one of the most important open problems in the theory of mapping class groups. It is a very hard question going back to the work of Dehn and Nielsen in the 1920s. Bestvina, Lucarelli, Vogtmann, and the PI are making significant progress by performing a computer-aided search. Various algorithms for Problem (3) are known, most notably the Bestvina-Handel algorithm. With Yurttas the PI has a new algorithm for computing train tracks that works in quadratic time; in practice it is much quicker than the Bestvina-Handel algorithm (which we conjecture to be doubly exponential). All three projects address fundamental questions in the theory of mapping class groups and in all three cases the PI and his collaborators have already made significant headway. In addition to these research goals, the PI also proposes to continue work on two major projects that have direct impact on graduate and undergraduate students. The first is the Topology Students Workshop, a conference that serves both as a research conference in topology for graduate students as well as a professional development workshop. The second is Office Hours with a Geometric Group Theorist, an introductory text on Geometric Group Theory for undergraduates.