Geometry/Topology Seminar

Winter 2009

Thursdays (and sometimes Tuesdays) 2-3pm, in Eckhart 308

- Thursday January 15 at 2pm in E308
- Mario Bonk, University of Michigan
- Quasiconformal rigidity of Sierpinski carpets
- Abstract: Sierpinski carpets are topologically very “flexible" objects. For example, the homeomorphism group of a carpet acts n-transitively on its collection of peripheral circles for every n. In contrast, surprising rigidity phenomena emerge if one considers quasisymmetric self-homeomorphisms of carpets. This is related to questions in geometric group theory. I will give a survey of recent developments in this area.
- Double Header - Part I
- Thursday January 22 at 2pm in E308
- Jason DeBlois, University of Illinois at Chicago
- Some virtually special 3-manifold groups
- Abstract: Haglund-Wise recently defined combinatorial critera by which a cube complex may be “special" and established a host of properties of special cube complexes. I will describe a class of hyperbolic 3-manifold groups with associated cube complexes which are virtually special, and discuss implications for the geometry and topology of such manifolds. This is joint work with Eric Chesebro and Henry Wilton.
- Double Header - Part II
- Thursday January 22 at 3pm in E308
- Matthew Day, Caltech
- Crossed homomorphisms on the mapping class group via nilpotent homogeneous spaces
- Abstract: I will explain how to use nilpotent homogeneous spaces to construct crossed homomorphisms (group cohomology 1-cocycles) on mapping class groups that extend the higher Morita homomorphisms, and crossed homomorphisms on automorphism groups of free groups that extend the higher Johnson homomorphisms. This improves my earlier results by restricting the images of these maps to finite-rank modules.
- Thursday January 29 at 2pm in E308
- Igor Pak, University of Minnesota
- Realization of polyhedral surfaces
- Abstract: I will survey both geometric and computational aspects of the following problem: given an abstract polyhedral surface, can one isometrically realize it in the Euclidean space? I will start with the classical Alexandrov and Zalgaller existence theorems and then outline recent developments, in part motivated by the discrete geodesic problem.
- Thursday February 5 at 2pm in E308
- Igor Rivin, Temple University
- Asymptotic geometry of convex sets in hyperbolic space
- Abstract: Motivated by an attempt to generalize Dvoretzky's theorem (which says that convex sets have almost round crosssections of large dimension) to the hyperbolic setting, we study how convex sets look at (and near) the ideal boundary of hyperbolic space. The talk will be more-or-less self-contained.
- Thursday February 12 at 2pm in E308
- Ben McReynolds, University of Chicago
- Controlling covers of arithmetic orbifolds
- Thursday February 26 at 2pm in E308
- Jeremie Brieussel, Cornell University
- Amenability of directed groups of rooted tree automorphism
- Abstract: I will show that the group of directed automorphisms of a rooted tree is amenable iff the valency of the tree is bounded. This result provides examples of amenable groups of non uniform growth.
- Thursday March 5 at 2pm in E308
- Yael Algom Kfir, University of Utah
- Negative Curvature phenomena in Outer Space
- Abstract: In a negatively curved space, projections to geodesics are strongly contracting: if we take a ball disjoint from the geodesic and project it to the geodesic then the diameter of the image is bounded independently of radius of the ball. We will discuss Culler and Vogtmann's Outer Space endowed with the Lipschitz metric, a space which admits a nice action of Out(F_n). We will then prove that axes of fully irreducible automorphisms admit strongly contracting projections. This can be applied to show that axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are Morse geodesics.
- Thursday March 12 at 2pm in E308
- Uri Bader, Technion
- Rigidity of centralizers for lattice actions
- Abstract: Consider the action of GL_n(Z) on Rⁿ. Clearly it commutes with homotheties. One may ask whether the group of homotheties consists of the full centralizer group of that action. More precisely: Given a measurable self map of Rⁿ (almost everywhere defined) which commute with every invertible integral matrix, need that be a multiplication by a scalar?
  In my talk I will explain how to affirmatively answer the question above, and, indeed, how to prove a much more general statement. The main ingredients of the proof are
  1. A certain Frobenius reciprocity relating suitable induction and restriction functors, and
  2. The Borel density theorem, essentially describing all invariant measures on an algebraic varieties.
  Both will be clearly explained during the talk. The talk is based on a joint work with Furman, Gorodnik and Weiss.
- Thursday March 19 at 2pm in E308
- Kenneth Shackleton, University of Tokyo IMPU
- Geodesics in the pants graph
- Thursday April 2 at 2pm in E308
- Bruno Klingler, Institut de Mathématiques de Jussieu
- On local rigidity for complex hyperbolic lattices and non-Abelian Hodge theory
- Abstract: One central problem in the study of lattices in simple real Lie groups is certainly the structure of complex hyperbolic lattices and their finite dimensional representations. Except a few examples the only known representations of such a lattice are those obtained by restricting representations of the ambient group SU(n,1) (n>1). Thus let L be a (cocompact) lattice and r: SU(n,1) -> G be a morphism into a real simple Lie group. Can we obtain new representations of L by deforming the restriction of r to L? In the 90's Goldman and Millson proved this is not possible for the standard embedding of SU(n,1) in SU(n+1,1). I will explain how non-Abelian Hodge theory provides a general answer to this problem.
- Thursday April 9 at 2pm in E308
- Jason Manning, University of Buffalo
- Geometric Dehn filling of high dimensional hyperbolic manifolds
- Abstract: We describe geometric methods for understanding the group theoretic Dehn fillings of a high dimensional cusped hyperbolic manifold. Such fillings are shown to act geometrically on either CAT(-1) spaces or CAT(0) spaces with isolated flats depending on the type of filling performed. The shape of the boundary is described and group theoretic information is inferred. This is joint work with Koji Fujiwara.
- Thursday April 30 at 2pm in E308
- Alan Reid, University of Texas

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Ben McReynolds, dmcreyn at math.uchicago.edu
Nathan Broaddus, broaddus at math.uchicago.edu