Topology Festival

May 8–10, 2015

Abstracts of Talks

Boundaries of Hyperbolic Spaces (Friday Introductory Talk)

Gromov defined the boundary of any $\delta$-hyperbolic space $X$. I will consider some basic examples and the relation between its dimension and the properties of $X$ or the group acting on it. If time permits, I will describe Klarreich's work on the boundary of the curve complex.

Boundaries of Some $Out(F_n)$-Complexes

Recently it has been shown that $Out(F_n)$ acts on several natural hyperbolic complexes. I will talk about the recent work with Reynolds on the boundary of the complex of free factors and about the work in progress with Feighn and Reynolds on the boundary of the complex of free splittings.

Overtwisted Contact Structures

The notion of an overtwisted contact 3-manifold and its associated h-principle has been a central part of contact topology for the last twenty-five years. Recently, in joint work with Y. Eliashberg and E. Murphy, we generalized the notion of an overtwisted contact structure to higher dimensions and proved the corresponding h-principal. In this talk I will survey this result and its place within the recent discovery of new flexibility phenomenon in contact topology.

Fixed Point Properties and Proper Actions on Non-positively Curved Spaces and on Banach Spaces

One way of understanding groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Classical properties like Kazhdan property (T) and the Haagerup property are formulated in terms of such actions and turn out to be relevant in a wide range of areas, from the construction of expanders to the Baum-Connes conjecture.

In this talk I shall overview various generalisations of property (T) and Haagerup to Banach spaces, especially in connection with classes of groups acting on non-positively curved spaces.

The Roller Boundary and CAT(0) Cube Complexes

The Roller Boundary of a CAT(0) cube complex $X$ is a natural compactification arising from the edge metric. It inherits a median structure from $X$ making it a beautiful and useful object of study. In this talk we will discuss its role in the super-rigidity results of Chatterji-Fernos-Iozzi and its connection to random walks on groups that admit nice actions on such $X$. More specifically, we will discuss how, under some standard assumptions, the Roller Boundary carries a stationary measure making it the Furstenberg-Poisson Boundary of a random walk on the acting group $G$.

Geometry of Contracting Geodesics (Friday Introductory Talk)

One coarse feature of a hyperbolic space is that geodesics are contracting. That is if $L$ is any geodesic then there exists $B$ such that the nearest point projection to $L$ of any metric ball disjoint from $L$ has diameter at most $B$. We will discuss a space with some but maybe not all geodesics are contracting, and its application to group theory. An example is a Teichmuller space. This is a joint work with Bestvina and Bromberg.

Handlebody Subgroups In A Mapping Class Group

Suppose subgroups $A, B < MCG(S)$ are given and let $\langle A,B\rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A,B\rangle = A*_{A \cap B} B$ for handlebody subgroups $A, B$. We construct an example such that Heegaard distance between $A$ and $B$ is arbitrarily large, $A \cap B$ is trivial but $\langle A,B\rangle$ is not $A*B$. This is a joint work with Bestvina.

Universal Groups of Prees

Gersten and Short have shown that groups satisfying the usual non-metric small cancellation conditions are biautomatic. In this talk we generalize the C(3)-T(6) condition to obtain a larger collection of biautomatic groups, and we also apply our technique to triangles of groups.

Parabolic Blowups

A polynomial $p$ of degree $d$ has a filled in Julia set
$$K_p = \{z\in \mathbb{C}\mid \text{the sequence}\ z, p(z), p(p(z)),\dots \text{ is bounded}\}$$ This set is a compact subset of $\mathbb{C}$, and does not depend continuously on $p$, specifically if $p$ has a parabolic cycle, i.e., if there is a point $z\in K_p$ such that $p^k(z) = z$ and $(p^{\circ k})'(z)$ is a root of unity. So it makes sense to ask what the closure of the set of all $K_p$ is in the “hyperspace” of all compact subsets of $\mathbb{C}$ for the Hausdorff metric.

This is analogous to asking what the closure of the set of Kleinian groups isomorphic to $\pi_1(S)$ is for some surface $S$. Thurston studied this under the name of geometric limits of hyperbolic 3-manifolds.

I will describe this space, as a complicated projective limit, that is however understandale in the sense that one can compute its Čech cohomology.

Highly Transitive Actions, Mixed Identities, and Acylindrical Hyperbolicity

The transitivity degree of a group is the supremum of transitivity degrees of its faithful permutation representations. This notion is classical and rather well-understood in the theory of finite groups. For infinite groups, however, very little is known. In my talk, based on a joint work with M. Hull, I will show that in some natural algebraic and geometric settings, the transitivity degree of a group can only take two values, namely $1$ and $\infty$. The crucial fact underlying this phenomenon is the following theorem of independent interest: Every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. Further, for any countable group $G$ admitting a highly transitive faithful action, we prove the following dichotomy: Either $G$ contains a normal subgroup isomorphic to the infinite alternating group or $G$ resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups.

High Dimensional Expanders

Recent years have seen several attempts to generalize the theory of expander graphs to higher dimensions. I will describe an approach which is inspired by Riemannian Hodge theory, relating strong vanishing of the $i$-dimensional cohomology to combinatorial expansion and pseudo-randomness. If time allows, I will discuss examples such as random and Ramanujan complexes, and comment on relations to other notions of expansion: Linial-Meshulam coboundary expansion, and Gromov's overlap properties. Based on joint works with Konstantin Golubev, Ron Rosenthal and Ran Tessler.

The Topology of Toric Origami Manifolds

The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of smooth generalizations of toric symplectic manifolds and projective toric varieties. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, but this degeneracy is enough to allow for non-simply-connected and non-orientable manifolds, which are excluded from the topological generalizations mentioned above.

In this talk we will see how the topology of an (orientable) toric origami manifold, in particular its fundamental group, can be read from the polytope-like object that represents its orbit space. These results are from joint work with Tara Holm.