Geometry/Topology Seminar

Thursdays at 3:00PM, in 308 Eckhart.
if you have problems viewing the abstracts click on the link Geometry/Topology Seminar Web Page

Fall 2004

September 16,
Mark Sapir , Vanderbilt
Asymptotic cones of relatively hyperbolic groups and quasi-isometry rigidity
Abstract: This is a joint work with Cornelia Drutu and Denis Osin. We give a characterization of strongly relatively hyperbolic groups in terms of their asymptotic cones. This allows us to show that many classes of relatively hyperbolic groups are closed under quasi-isometries. We also prove the rapid decay property for relatively hyperbolic groups whose parabolic subgroups have these property and answer a question of Gromov about the possible number of asymptotic cones of a group.

October 7,
Ilya Kapovich , University of Illinois, Urbana-Champaign
Kleinian groups and the rank problem
Abstract: We prove that the rank poblem (that is, computing the smallest cardinality of a generating set) is solvable for the class of word-hyperbolic torsion-free Kleinian groups. We also show that every group of this class has only finitely many Nielsen equivalence classes of generating sets of a given size. This is joint work with Richard Weidman.

October 14,
Ilya Kapovich , University of Illinois, Urbana-Champaign
Translation equivalence in free groups

October 19, (Special Day)
Danny Calegari , CalTech
Low-dimensional dynamics I

October 21,
Mark Sapir , Vanderbilt
Residually finite groups and dynamics of polynomial maps over finite fields
Abstract: The talk is based on joint work with Alexander Borisov. We show that all ascending HNN extensions of free groups are residually finite. This implies that most 1-related groups are residually finite. The key part of the proof is counting the number of quasi-fixed points of polynomial maps of algebraic varieties over finite fields (we prove a version of a Deligne conjecture). We shall also discuss linearity of these groups, applications to extensions of endomorphism of free groups to their pro-finite completions, connections with random walks on Z^2, etc. In particular, we give the first example of non-linear residually finite 1-related group.
October 27, (Special day)
Azer Akhmedov , UC Santa Barbara
Quasi-Isometric Rigidity in Group Varieties
Abstract: We prove that the property of satisfying a law (i.e. belonging to a group variety) is not preserved under quasi-isometry. The construction is heavily based on the study of newly introduced travelling salesman property in finitely generated groups. Same methods allow to build some other counterxamples.
October 28, (Date Changed)
Danny Calegari CalTech
Low dimensional dynamics II
Tuesday November 2 (Special Day)
Vincent Lafforgue ENS Paris
Strong Property (T)
Abstract: We say that a loccaly compact group has a strong property (T) if the trivial representation is isolated among representations in Hilbert spaces, which are non neccesary unitary, but with polynomial and even small exponential growth. We prove that SL(3, R) adn SL(3, Q_p) have this strong property (T) but hyperbolic groups do not have it.

November 4,
Lisa Carbone , Rutgers
Lattices, Buildings and Kac-Moody groups
Abstract: Let K be a Kac-Moody Lie algebra and let G denote a Kac-Moody group associated with K. When K is of affine type, the commutator subalgebra of K is a central extension of a loop algebra and G is a central extension of a classical group. However if K is of hyperbolic type, many fundamental questions about the structure of K and G remain unanswered. Using representation theory of K and working over finite fields, Carbone and Garland have developed structure theorems for hyperbolic Kac-Moody groups G and their lattices. Here we show that although G has no obvious arithmetic or algebraic structure, lattices in rank 2 Kac-Moody groups contain congruence subgroups. These are defined in direct analogy with congruence subgroups of arithmetic lattices in simple algebraic groups.
November 9
Brian Hall Notre Dame
Analytic continuation and quantization for symmetric spaces
Abstract: I will describe something called the Segal-Bargmann transform for symmetric spaces, such as spheres and hyperbolic spaces. The transform consists of taking a function on the symmetric space, applying the heat operator, and then analytically continuing to an appropriate "complexification" of the symmetric space. In the compact case (e.g. a sphere), this is by now well understood and yields a unitary map from L^2 of the symmetric space to a certain L^2 space of holomorphic functions on the complexification. In the noncompact case (e.g. hyperbolic space), the situation is much more subtle because of singularities that arise in the analytic continuation. Nevertheless, recent progress has been made. I will begin with a discussion of the Euclidean case (i.e. R^n) and the compact case, and then describe some recent results (with J. Mitchell) on the noncompact case.
Wednesday, November 10, (Special Date) 3pm at Eckart 308
Danny Calegari CalTech
quasigeodesic flows

November 11,
Alexander Buffetov , Princeton University
Abstract: The talk will be devoted to the proof of a stretched exponential bound for the decay of correlations of the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A corollary is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials.

November 18,
Thomas Foertsch , University of Michigan
Curvature bounds in coarse geometry" (joint work with Mario Bonk)
Abstract:In coarse geometry one is interested in the geometric features of a space on large scales. In my talk I will show that in this context one can develop a theory of spaces with curvature bounded from above. This is joint work with Mario Bonk.

November 23,
Tsachik Gelender , Yale University
Classification of infinite pimitive groups
Abstract: I'll describe a joint work with Yair Glasner. We investigate infinite, finitely generated groups that admit a faithful primitive action on a set. We can give a complete characterization of such groups in different settings, including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The later category includes as a special case Kleinian groups as well as finitely generated subgroups of word hyperbolic groups. As an application we calculate the Frattini group in a few of these settings. In particular we settle a conjecture of Higman and Neumann on the Frattini group of an amalgamated free product. We also prove that the Frattini group of a convegence group is finite, generalizing a theorem of Kapovich.

December 2,
R. Brown , American University
The dynamics of topology on the geometry of a surface
Abstract: Topological equivalence is rarely respectful of geometry. In this talk, I will discuss the dynamics of how the topological transformations of a surface affect geometric structures defined on that surface. Specifically, we will study how the mapping class group of the surface acts on the spaces of geometric structures modeled on the representation varieties of the surface's fundamental group (a way to characterize some moduli spaces associated to the surface). This action reveals properties of both the moduli spaces and the acting mapping class group.