Topology Festival image

Topology Festival

May 7–10, 2010

Abstracts of Talks

Miklos Abert, University of Chicago

Graph Limits, Covering Towers, and the Dynamics of Profinite Actions

There are some natural invariants of graphs, groups and manifolds where the asymptotic behavior can be productively studied using analytic tools. I will survey on the known results and directions and state many open problems.

Indira Chatterji, Ohio State University and Université d'Orléans

Subgroup Distortion and Bounded Cohomology

We discuss subgroup distortion and explain why, given a connected Lie group G, its fundamental group is undistorted in its universal cover if and only if the radical of G is linear. This is joint work with Ch. Pittet, G. Mislin, and L. Saloff-Coste.

Karsten Grove, University of Notre Dame

Positive Curvature in the Presence of Symmetries

There have recently been remarkable advances in the classical area surrounding manifolds with positive curvature. Part of this is due to investigations relating to the presence of large groups of isometries. This has led to classification type results as well as to the discovery of examples.

Jeremy Kahn, Stony Brook University

Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds

Given any closed hyperbolic 3-manifold M and ε > 0, we find a closed hyperbolic surface S and a map f : S → M such that f lifts to a (1 + ε)-quasi-isometry from the universal cover of S to the universal cover of M. It follows that, for ε small, the map f induces an injection on the fudamental group of S; thus the fundamental group of every closed hyperbolic 3-manifold has a surface subgroup. This is joint work with Vladimir Markovic of Warwick University.

I will explain why the mixing of the frame flow on M implies the existence of a highly symmetric collection of pairs of pants, which can then be assembled to form the desired surface S.

Dan Margalit, Tufts University

Problems and Progress on Torelli Groups

Torelli groups were first studied by Nielsen and Magnus in the early twentieth century, and subsequently fell out of fashion for fifty years. In the past three decades, the study of the Torelli group has gone from being a fringe topic, even within surface topology, to a deep, rich, and exciting theory connecting many different areas of mathematics. We will focus on the group theoretical and homological properties of these groups, and our talk will span the history of this topic, from the foundational work of Joan Birman and Dennis Johnson on generating sets for Torelli groups, to recent work of the speaker on cohomological properties, which is joint with Bestvina, Bux, Brendle, and Putman. We will emphasize the key constructions, as well as open questions and conjectures.

Nikolay Nikolov, Imperial College London

Rank Gradient of Groups and Applications

Rank gradient is a tool first introduced by Marc Lackenby for the study of fundamental groups of hyperbolic 3-manifolds. Since then this purely group theoretic notion has turned out to have applications to several different mathematical areas: measurable group actions, graph theory, arithmetic groups and geometry. In this talk I will explain these connections and survey the main problems, which are still wide open.

This is joint work with Miklos Abert.

Doug Ravenel, University of Rochester

The Arf-Kervaire Invariant Problem

Mike Hill, Mike Hopkins, and I recently solved the 50-year-old Arf-Kervaire invariant problem in algebraic topology. The talk will describe the background and history of the problem and give a brief overview of the proof of our main theorem. More information can be found at

Ed Swartz, Cornell University

Counting Faces Since Poincaré

It has been over 100 years since Poincaré’s original papers on the topological invariance of the Euler characteristic. What have we learned about the face numbers of triangulated spaces since then? The answer is astonishingly little. For instance, there is not even a credible conjecture which characterizes the face numbers of simplicial balls of dimension six or higher. We will survey various aspects of what has been discovered since Poincaré. Along the way we will see how the introduction of commutative algebra revolutionized the subject in the 70’s, and several new results concerning manifolds and pseudomanifolds.

Daniel Wise, McGill University

The Structure of Groups with a Quasiconvex Hierarchy

We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on “special cube complexes” which are nonpositively curved cube complexes that behave like “high dimensional graphs” and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is “virtually special,” and this yields the separability of the quasiconvex subgroups of the groups we study.

As an application, we resolve Baumslag’s conjecture on the residual finiteness of one-relator groups with torsion. Another application shows that generic Haken hyperbolic 3-manifolds have “virtually special” fundamental group. Since graph groups are residually finite rational solvable, combined with Agol’s virtual fibering criterion, this proves that finite volume Haken hyperbolic 3-manifolds are virtually fibered.

Robert Young, IHES

The Dehn Function of SL(n;Z)

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is SL(n;Z). Thurston conjectured that SL(n;Z) has a quadratic Dehn function when n ≥ 4. This differs from the behavior for n = 2 (when the Dehn function is linear) and for n = 3 (when it is exponential). I have proved Thurston’s conjecture when n ≥ 5, and in this talk, I will give an introduction to the Dehn function, discuss some of the background of the problem and give a sketch of the proof.