Schedule of the Workshop: Subgroups of mapping class groups

Shinpei Baba: 2π-graftings on complex projective structures
A (complex) projective structure is a certain geometric structure on a (closed) surface, that is a natural generalization of hyperbolic structure. Then each projective structure corresponds to a (holonomy) representation of the surface group into PSL(2,C). However, this correspondence is not one-to-one. (2p-)grafting is a certain surgery operation on a projective structure that produces a different projective structure with the same representation. We show that, given two projective structures with the same generic representation, there is a sequence of graftings and inverse-graftings that transforms one to the other. (2p-)grafting is a certain surgery operation on a projective structure that produces a different projective structure with the same representation. We show that, given two projective structures with the same generic representation, there is a sequence of graftings and inverse-graftings that transforms.

Richard Canary: Moduli spaces of hyperbolic 3-manifolds
It is common to study the moduli space of closed hyperbolic surfaces of a fixed genus. In 3 dimensions, one usually studies the space AH(M) of (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3- manifold M. One may regard AH(M) as the 3-dimensional analogue of the Teichmüller space of marked hyperbolic surfaces of a fixed genus. In this talk, we discuss the moduli space of hyperbolic 3-manifolds homotopy equivalent to M, which is the quotient of AH(M) by the action of the outer automorphism group of the fundamental group of M. We will describe results on the topology of the moduli space (which can be quite pathological) and the dynamics of the action of the outer automorphism group on AH(M) and more generally on the PSL(2,C)-character variety of M. This talk describes joint work with Pete Storm.

Vaibhav Gadre: Hitting measures on PMF
Kaimanovich and Masur showed that for a random walk on the mapping class group G, almost every sample path converges to a point in the space PMF of projective measured foliations. In particular, one gets a well-defined hitting measure on PMF. In this talk, we shall consider random walks on G that arise from finitely supported initial distributions. We shall explain why the hitting measures for such random walks are singular with respect to the natural Lebesgue measure on PMF.

Christopher Leininger: Subgroups of the mapping class group from a geometrical viewpoint
We will discuss some of the many analogies between subgroups of the mapping class group and Kleinian groups. Much of this talk is based on joint work with Richard Kent.

Samuel Lelievre: The geometry of spheres in free abelian groups
Call sprawl of a group (with respect to a generating set) the limit, if it exists, of 1/n times the average distance in the word metric between pairs of words of length n. This invariant quantifies a certain obstruction to hyperbolicity. For free abelian groups, we show that counting measure on spheres in any word metric converges to cone measure on a convex polyhedron, and use this to reduce the averages of asymptotically homogeneous functions to a computation in convex geometry via a counting technique that has a number of independent applications, including estimates for growth functions and spherical growth functions. We present an algorithm for computing the sprawl and some results about its values, including a connection to the Mahler conjecture. This is joint work with Moon Duchin and Christopher Mooney.

David Ben McReynolds: Congruence subgroups, holomorphic curves, and breaking symmetry
I will discuss the congruence subgroup problem for a class of groups, including mapping class groups, and give a brief survey of the status of the problem. I will discuss algebraic and geometric symmetry problems that arise in addressing this problem and ties with problems in Riemannian geometry. Finally, I will discuss a general method for producing holomorphic curves in moduli spaces with an emphasis on Teichmuller curves. This is joint work with Jordan Ellenberg.

David Ben McReynolds: Every curve is a Teichmuller curve
I will give a detailed account of recent work with Jordan Ellenberg. The main goal is to prove a pair of results. First, that every algebraic curve defined over the Galois closure of the rationals is birational to a Teichmuller curve. Second, every finite index subgroup of SL(2,Z), which is contained in the level two congruence kernel and contains the center, is a Veech group. The base idea is geometric (and given in my first talk) but our proofs are algebraic. I will sketch the main steps in proving the second result as by Belyi’s Theorem, it implies the first.

Piotr Przytycki: The ending lamination space for the five-punctured sphere

We prove that the boundary of the curve complex for the 5-punctured sphere is the so called Noebeling curve. This is a space obtained from R3 by removing all points with at least 2 rational coordinates.

Saul Schleimer: The graph of handlebodies
(Joint work with Joseph Maher.) We introduce the graph of handlebodies and prove that it is quasi-isometric to an electrification of the curve complex. We show that this graph is Gromov-hyperbolic and of infinite diameter.