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Topology Festival

May 21–24, 2007

Abstracts of Talks

Danny Calegari, California Institute of Technology

Curvature and Stable Commutator Length

Stable commutator length, or "rational filling genus," is a natural function on the commutator subgroup [G,G] of a group G. It has connections with (second) bounded cohomology, and is currently a topic of some interest in geometry and dynamics. We discuss some surprising properties of this function in the case when G is (coarsely) negatively curved or non-positively curved. Among the properties we discuss are

  1. spectral gap;
  2. rationality;
  3. expected growth rate
Some of this work is joint with Koji Fujiwara.

Ralph Cohen, Stanford University

Surfaces in a Background Manifold and the Homology of Mapping Class Groups

In this talk I will describe joint work with Ib Madsen in which we study the topology of the space of Riemann surfaces in a simply connected manifold X, Sg,n(X, γ). This is the space consisting of pairs, (Fg,n, f ), where Fg,n is a Riemann surface of genus g and n-boundary components, and f : Fg,nX is a smooth map that satisfies a boundary condition γ. Our main theorem is the identification of the stable homology type of the space S∞,n(X, γ), defined to be the limit as the genus g gets large, of the spaces, Sg,n(X, γ). Our second result describes a stable range in which the homology of Sg,n(X, γ) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are extensions of stability theorems of Harer and Ivanov.

Cornelia Drutu, Université des Sciences et Technologies de Lille I

Relatively Hyperbolic Groups: Geometry and Quasi-isometric Invariance

The topic of the talk is quasi-isometry invariance of relative hyperbolicity (with and without preservation of classes of quasi-isometry of peripheral groups). Important ingredients in the proofs of such results are some simplified definitions of (strong) relative hyperbolicity in terms of the geometry of a Cayley graph. In particular one definition is very similar to the one of hyperbolicity, as it relies on the existence for every quasi-geodesic triangle of a central left coset of peripheral subgroup. Part of the results presented are from joint work with M. Sapir, and with J. Behrstock and L. Mosher.

Alex Eskin, University of Chicago

Counting Problems in Teichmüller Space

We apply ideas from the Ph.D. thesis of G.A. Margulis to Teichmüller space, and prove asymptotic formulas as R goes to infinity for for the number of closed geodesics of length at most R, the volume of ball of radius R, and the number of lattice points in a ball of radius R.

Parts of this talk are joint work with Maryam Mirzakhani. Other parts are joint work with Jayadev Athreya, Sasha Bufetov and Maryam Mirzakhani.

Mark Feighn, Rutgers University at Newark

Definable Subsets of Free Groups

I will describe the current state of our project to understand the structure of definable subsets of a free group F, i.e. sets consisting of free group elements satisfying an open sentence in the first order theory of F.

Our methods, inspired by Sela's work on the Tarski problem, are geometric. Definable subsets of F are reinterpreted in terms of homomorphisms from groups into F. If we fix a Cayley graph for F, actions of groups on simplicial trees arise. The structure of a definable set is probed by considering the real trees that arise as limits of sequences of these simplicial trees.

Ilya Kapovich, University of Illinois at Urbana-Champaign

Geodesic Currents and Outer Space

We will discuss the properties of the intersection form, similar to Bonahon's notion of an intesection number between two geodesic currents on a hyperbolic surface, between the outer space and the space of geodesic currents on a free group. In particular, we will consider the notions of "filling" currents and "filling" conjugacy classes, with applications to bounded translation equivalence in free groups. We will also investigate discontinuity domains for the actions of subgroups of Out(Fn) on the boundary of the outer space and on the space of projectivized currents.

Chris Leininger, University of Illinois at Urbana-Champaign

The Boundary of the Curve Complex

According to a theorem of Masur and Minsky, the curve complex of a surface is hyperbolic in the sense of Gromov. Klarreich then proved that the Gromov boundary is naturally homeomorphic to a quotient of the space of arational measured foliations (a subspace of the space of all measured foliations). The curve complex and its boundary have played an important role in the proof of Thurston's ending lamination conjecture, and are proving to be of particular interest in the study of the mapping class group. In this talk I'll explain why the space of arational measured foliations — and hence also the boundary of the curve complex — is usually connected. This is joint work with S. Schleimer.

Tim Riley, Cornell University

The Geometry of Discs Spanning Loops in Groups and Spaces

Geometric features of discs spanning loops in spaces are recorded by invariants known as filling functions. Isoperimetric functions, which concern the infimal area of discs spanning a given loop, are the classical example. Filling functions of a (finitely presentable) group are defined via a space (e.g. Riemannian manifold) on which the group acts in a suitably nice way. They provide means of elucidating large-scale geometry and are quasi-isometry invariants. I will discuss how different geometric features of discs, such as their diameter, their area, and their "filling length," interact in this setting. I will explain what this means for interrelationships between the associated filling functions.

Juan Souto, University of Chicago

Heegaard Splittings and Hyperbolic Geometry

I will discuss recent progress towards understanding how the geometry of hyperbolic 3-manifolds is encoded in combinatorial information provided by a Heegaard splitting. This is joint work with Jeff Brock, Hossein Namazi and Yair Minsky.

Gang Tian, Princeton University

Geometrization of Low Dimensional Manifolds

In this talk, I will give a brief tour of Perelman's proof for the geometrization of 3-manifolds by using Hamilton's Ricci flow. I will also discuss some geometric aspects of 4-manifolds in the end. This talk is aimed at a general audience.