Tarik Aougab

Characterizing super-linear divergence and subgroup stability

A finitely generated subgroup H of a finitely generated group G is stable in G if it satisfies a certain strong form of quasi-convexity, which is a natural generalization of the notion of convex cocompactness for Kleinian groups or for subgroups of the mapping class group. We give new characterizations of stability in terms of the divergence function. We also give a new characterization of super-linear divergence for a geodesic and a new proof that it is equivalent to Morse stability that does not use asymptotic cone machinery. Using these and related results, we give a sufficient condition for subgroup stability that we use to identify stable subgroups of (1) the outer automorphism group of a free group and of (2) relatively hyperbolic groups. This represents work in progress with Matthew Durham and Samuel Taylor.

David Cohen

Lipschitz 1-connectedness for some solvable groups

In a finitely presented group, every relation can be written as a product of conjugates of relators. The Dehn function of the group measures the number of conjugates of relators needed for a relation of length n. Equivalently, it measures the area needed to fill a loop of length n in the universal cover of the presentation complex of the group. A group is said to be Lipschitz 1-connected if every length n loop in the universal cover of its presentation complex has a O(n)-Lipschitz filling over the unit disk. Lipschitz 1-connectedness implies quadratic Dehn function, but it is not known whether the converse is true. A result of Cornulier and Tessera gives a quadratic Dehn function for many solvable groups. We show that many of these groups are Lipschitz 1-connected.

Asaf Hadari

Homological representations of mapping class groups

Our understanding of the representation theory of mapping class groups is in a somewhat unusual state. On the one hand, we know that they have a large collection interesting representations. On the other hand, we don't know the answers to some incredibly basic questions about these representations. I will discuss the resolution of one such question: Given an infinite order mapping class, is there some homological representation where its image has infinite order?

Thomas Koberda

Unsmoothable group actions on one-manifolds

I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful C2 action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod.

Kathryn Mann

Linear and circular orders on groups

A nice property of the integers (as an additive group) is that the usual order relation < is preserved by the group operation, i.e. "m < n implies a+m < a+n". Surprisingly, quite a number of other groups also admit such a total order invariant under left multiplication. These "left-orders" can be characterized algebraically, but often arise from geometric and topological ideas, and in turn can give us valuable dynamical or topological information. This talk will introduce you to left-orderable groups — and their close cousins, circularly orderable groups — through examples, exercises, and applications (and maybe some open questions).

Spaces of linear and circular orders on groups

Although a left-invariant linear or circular order on a group is an algebraic condition, the question of existence and classification of such orders is deeply connected to problems in geometry and topology (e.g. existence of foliations) and dynamics (group actions on manifolds). In this talk, I'll present new perspectives on the space of all circular orders on a group G, and its relationship with the space of actions of G on the circle. As a consequence, I'll give a dynamical characterization of “isolated circular orders”, discuss some rigidity phenomena, and give a proof that free groups admit many isolated circular orders. This is joint work with Cristobal Rivas.

Denis Osin

Group theoretic Dehn surgery

Group theoretic Dehn surgery is an algebraic generalization of Thurston's theory of hyperbolic Dehn filling in 3-manifolds. It turned out to be useful in proving a wide range of results, from purely group theoretic theorems to the Farrell-Jones conjecture for relatively hyperbolic groups to the virtual Haken conjecture for hyperbolic 3-manifolds. The purpose of my talk will be to give an elementary introduction to this theory and to survey some recent generalizations and applications.

Transitivity degrees of countable groups

The transitivity degree of a group is the supremum of transitivity degrees of its faithful permutation representations. This notion is classical and well-understood for finite groups. For infinite groups, however, very little is known and many basic questions are open. I will explain how to answer these questions for various groups of geometric origin.

Piotr Przytycki

Arcs intersecting at most once

Consider a punctured surface with negative Euler characteristic X. We prove that the maximal cardinality of a collection of arcs that are pairwise non-homotopic and intersecting at most once is 2|X|(|X|+1).

Arcs intersecting at most twice

This is joint work with Christopher Smith. We prove that on a punctured sphere the maximal cardinality of a collection of arcs that are pairwise non-homotopic and intersecting at most twice is |X|(|X|+1)(|X|+2).

Robert Young

Quantifying nonorientability and filling multiples of embedded curves

Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in R^n which is only about 1.5 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. We will connect this question to a way of measuring the nonorientability of a manifold in R^n and answer it using methods from geometric measure theory.