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.
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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.
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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.
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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.
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Wednesday, November 10, (Special Date) 3pm at Eckart 308
Danny Calegari CalTech
quasigeodesic flows
November 11,
Alexander Buffetov , Princeton University
DECAY OF CORRELATIONS FOR THE RAUZY-VEECH-ZORICH INDUCTION MAP
AND THE CENTRAL LIMIT THEOREM FOR THE TEICHMUELLER FLOW
| 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.
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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.
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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.
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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.
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