Workshop on Geometric Group Theory and Geometric Topology

October 16 - 17, 2015

A related event: Virginia Mathematics Lecture Series by Ian Agol

Abstracts:

Jason Behrstock: Random graphs and applications to Coxeter groups

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then, explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

Matthew Durham: A new boundary for the mapping class group

I will introduce a geometrically intrinsic compactification of the mapping class group which, in the course of proving theorems, functions like the Gromov boundary of a hyperbolic group. I will discuss some initial applications, including a Rank-Rigidity type theorem and the fact that the boundary is a model for the Poisson boundary for random walks on the mapping class group. Time permitting, I will indicate some future directions. This is joint work with Mark Hagen and Alessandro Sisto.

Boris Lishak: Balanced finite presentations of the trivial group and geometry of four-dimensional manifolds.

We will construct a sequence of finite presentations of the trivial group with just two generators and two relators that can be tranformed to the trivial presentation only by means of an enormously large number of Tietze transformations. We will explain several applications of such results to 4-dimensional geometry. For example, one can show that the space of metrics on a closed 4-manifold of fixed volume of injectivity radius bounded from below is disconnected (for any small enough bound). Another application is the existence of non-trivial "thick" 4-knots in the 5-dimensional Euclidean space. One can also construct trivial 2-knots in the 4-sphere that are very difficult to untie. (These geometric applications are a joint work with A. Nabutovsky.)

Dan Margalit: Models for mapping class groups

A celebrated theorem of Nikolai Ivanov states that the automorphism group of the mapping class group is the mapping class group itself. The main ingredient is another theorem of Ivanov which states the the automorphism group of the complex of curves is the mapping class group. After many similar theorems were proven, Ivanov made a metaconjecture that any "sufficiently rich object" associated to a surface should have automorphism group isomorphic to the mapping class group. In joint work with Tara Brendle, we show that there is a very wide class of normal subgroups of the mapping class group that have automorphism group isomorphic to the mapping class group. Following Ivanov, the main ingredient is that there is a wide class of complexes of curves that have automorphism group isomorphic to the mapping class group.

Alex Suciu: Complex geometry and 3-dimensional topology

I will discuss some of the interplay between complex algebraic geometry and low-dimensional topology, as it occurs when studying the fundamental groups of smooth, quasi-projective varieties and 3-dimensional

manifolds, or the Milnor fibration of an arrangement of complex planes. The bridge between the two settings is provided by the Alexander polynomial and some of its generalizations.

Jing Tao: Effective quasimorphisms on right-angled Artin groups

I will discuss a new technique for studying group actions on CAT(0) cube complexes. Our main application is the construction of well-behaved quasimorphisms on right-angled Artin groups. This leads to a gap theorem for stable commutator length in these groups. This is joint work with Talia Fernos and Max Forester.

Samuel Taylor: Hyperbolic bundles: From 3-manifolds fibering over the circle to hyperbolic extensions of free groups

As part of his celebrated hyperbolization theorem, Thurston proved that a fibered 3-manifold M admits a hyperbolic structure if and only if M is the mapping torus of a pseudo-Anosov homeomorphism. Thinking in terms of the induced splitting of the fundamental group of M, Thurstonâ€™s theorem suggests the possibility of hyperbolizing more general group extensions that live beyond the world of 3-manifold topology.

In this talk I will discuss recent work developing the theory of hyperbolic extensions of free groups and highlight the surprising similarities these findings share with Thurston's original theory. In addition to surveying the geometry of these hyperbolic extensions, I will also present some new results which quantitatively describe how topological and combinatorial data is transferred between various fibrations of the 3-manifold M. This talk includes joint work with S. Dowdall and Y. Minsky.