The purpose of this project is to study the mapping class group and its subgroups by looking at their actions on the Teichmueller space and on certain simplicial complexes encoding the combinatorics of ”nice” topological objects on the surfaces (curves, arcs and triangulations). The main protagonists will be the curve complex, the arc complex and the flip graph of topological and geometric surfaces. The curve complex is an important tool in the proof of Thurston’s Ending Lamination Conjecture and in the study of the coarse geometry of the mapping class group. We will be particularly interested in investigating the large-scale geometry and the model-theoretical properties of these graphs and their variations for surfaces endowed with ”good” geometric structures. In this project we want to establish the first bridge between geometric topology, model theory, geometric group theory and dynamics.
This project consists of three parts:
I. combinatorial actions of the mapping class group and rigidity;
II. large scale geometry of complexes of (multi-)arcs and interactions between geometric group theory and dynamics;
III.Thurston’s distance on the Teichmueller space and its generalizations in higher Teichmueller theory.
Part I deals with mapping class groups and the simplicial rigidity problem. In the 1990s Ivanov proved that the mapping class group can be represented as the automorphism group of the curve complex. Subsequently, many other simplicial complexes associated to a surface have exhibited this same feature. Understanding what all these objects have in common is still an open problem (Ivanov’s metaconjecture). Recently Brendle-Margalit characterize one entire family of complexes with this property and formulated a precise conjecture in this regard. I am attacking this problem for multi-arcs and studying the model theory of the curve complex and other graphs associated to mapping class groups together with Thomas Koberda and Javier de la Nuez-Gonzalez.
Part II deals with the geometric group theory of arc complexes, flip graphs, related algorithmic problems and a new application of these tools in dynamics. After foundational works by Masur-Minsky there has been a lot of interest in the large scale properties of the curve complex and other analogue combinatorial complexes. A recent trend in geometric group theory is to look at complexes of arcs and triangulations. Further motivations come from the many applications these objects have in other fields of mathematics (theoretical computer science, cluster algebras, quantum topology, Margulis space times...). I am interested in understanding the large scale geometry of the mapping class group via graphs of triangulations. I am developing a program with Anja Randecker and Robert Tang to employ techniques from geometric group theory in dynamics via the study of simplicial complexes built from saddle connections on a translation surface.
Part III deals with Thurston’s distance on Teichmueller space. Thurston’s distance is an analogue of Teichmueller distance defined by Thurston. Together with Daniele Alessandrini we are generalizing Thurston’s results to surfaces with boundary.