Groups and Dynamics Seminar

Organizers: Rostislav Grigorchuk, Volodia Nekrashevych, and Zoran Šunić.

	Date   Time	Location	Speaker	Title – click for abstract
	02/01   3:00pm	MILN 317	Russ Thompson Texas A&M University	The rate of escape for random walks on some polycyclic and metabelian groups How fast a random walk on a group escapes from its initial position has become an important problem in recent years. I'll begin by surveying what is currently known about this topic and how it relates to other problems in geometric group theory. I'll then go into my own work on the subject, which shows that any simple symmetric random walk on a polycyclic group has the same rate of escape as a simple symmetric random walk on the integers so long as the Fitting subgroup has uniform exponential distortion. The ideas behind this proof are generalized to metabelian groups via an analogy to the abelian sandpile model.
	02/20   3:00pm	BLOC 120	Anatole Katok	Speaker: Dr. Anatole Katok `Chaos vs. uniform distributions: Are they compatible?"
	02/29   3:00pm	MILN 317	Zoran Sunic  Texas A&M University	Sigma invariants of some self-similar groups We recall the notion of a Sigma invariant (also known as Bieri-Neumann-Strebel invariant) and provide examples related to some self-similar groups.
	03/07   3:00pm	MILN 317	Hossein Namazi UT Austin	The density conjecture and deformations of hyperbolic structures We give an overview of some of the main results in the study of Kleinian groups that have been developed recently. Our aim will be to show how they fit together and prove what is well known as Bers' density conjecture.
	03/21   03:00am	MILN 317	Dmytro Savchuk State University of New York - Binghamton	On restricting free factors in relatively free groups Using mainly Fox Calculus techniques we prove the following. If G is a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n and A = {a_1,..., a_r} is a free basis, then, in most cases, if S is a subset of some basis for G which may be expressed as a word in A without using elements from {a_{l+1},...,a_r}, then S is a subset of a basis for the relatively free group on {a_1,...,a_l}. The question was initially motivated by studying the proposed curve complex analog for Out(F_n). We will start from a motivation, then introduce main tools required for the proof and survey the proof in most cases. This is a joint work with Lucas Sabalka.
	03/27   3:00pm	Milner 216	Andrew Putman Rice University	Small generating sets for the Torelli group Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space (the handle graph of a surface) on which the Torelli group acts cocompactly.
	04/11   3:00pm	MILN 317	Rostislav Grigorchuk Texas A&M	On topological full groups of minimal homeomorphisms of a Cantor set I will describe the main properties of the full topological groups associated with minimal homeomorphisms of a Cantor set and will explain why they can be locally embedded into finite groups (i.e. are LEF groups). This will be used to get the first examples of infinite finitely generated simple groups with property LEF. This is joint work with K.Medynets.
	04/18   3:00pm	MILN 317	John Meakin University of Nebraska	Idempotent generated semigroups and von Neumann regular rings There is a rich literature about semigroups generated by idempotents, dating back to the work of von Neumann who developed the notion of (von Neumann) regular rings to coordinatize continuous geometries. Such rings may be characterized as rings whose principal left [right] ideals are generated by idempotents. The principal left [right] ideals of a regular ring form anti-isomorphic complemented modular lattices that may be used to understand the structure of the biordered set of idempotents of the ring. In this talk I will discuss some of the literature on idempotent generated semigroups, with particular emphasis on the problem of understanding the structure of the maximal subgroups of semigroups freely generated by the biordered set of idempotents of the multiplicative semigroup of some regular ring R. Even in the case when R is the ring M_n(Q) of n by n matrices over a field or division ring Q, this problem remains unsolved. Such a group associated with an idempotent matrix of rank k < n may be presented as the fundamental group of a 2-complex naturally associated with the set D_k of matrices in M_n(Q) of rank k. The structure of these groups depends in an essential way on combinatorial properties of the structure matrix associated with D_k. For k=n-1 the group is a free group. For k < n/3, Gray and Dolinka have recently extended results of Brittenham, Margolis and Meakin to show that this group is isomorphic to the general linear group GLk (Q). Additional results about the structure of these groups and the combinatorial structure of D_k for k >= n/3 will also be discussed.
	04/25   3:00pm	MILN 317	Volodymyr Nekrashevych Texas A&M University	Locally connected Smale spaces A Smale space is a dynamical system such that a neighborhood of every point is naturally decomposed into a direct product of expanding and contracting dynamics. Important examples of Smale spaces are Anosov diffeomorphisms. No other examples of locally connected Smale space are known. Moreover, all known Anosov diffeomorphisms come from hyperbolic automorphisms of nilpotent Lie groups. I will talk about a generalization to Smale spaces of a result of M. Brin on Anosov diffeomorphisms with pinched spectrum. The spectral condition of M. Brin will be replaced by a weaker condition on exponential rates of divergence of points, defined in purely topological terms.
	05/02   3:00pm	MILN 317	Lewis Bowen Texas A&M University	Invariant Random Subgroups of Free Groups Let G be a locally compact group. A random closed subgroup with conjugation-invariant law is called an invariant random subgroup or IRS for short. We show that each nonabelian free group has a large ``zoo'' of IRS's. This contrasts with results of Stuck-Zimmer which show that there are no non-obvious IRS's of higher rank semisimple Lie groups with property (T).

Topics

GENERAL PROBLEMS Burnside Problem on torsion groups, Milnor Problem on growth, Kaplanski Problems on zero divisors, Kaplanski-Kadison Conjecture on Idempotents, and other famous problems of Algebra, Low-Dimensional Topology, and Analysis, which have algebraic roots.

GROUPS AND GROUP ACTIONS Group actions on trees and other geometric objects, lattices in Lie groups, fundamental groups of manifolds, and groups of automorphisms of various structures. The key is to view everything from different points of view: algebraic, combinatorial, geometric, and probabalistic.

RANDOMNESS Random walks on groups, statistics on groups, and statistical models of physics on groups and graphs (such as the Ising model and Dimer model).

COMBINATORICS Combinatorial properties of finitely-generated groups and the geometry of their Caley graphs and Schreier graphs.

GROUP BOUNDARIES Boundaries of finitely generated groups: Freidental, Poisson, Furstenberg, Gromov, Martin, etc., boundaries.

AUTOMATA Groups, semigroups, and finite (and infinite) automata. This includes the theory of formal languages, groups generated by finite automata, and automatic groups.

DYNAMICS Connections between group theory and dynamical systems (in particular the link between fractal groups and holomorphic dynamics, and between branch groups and substitutional dynamical systems).

FRACTALS Fractal mathematics, related to self-similar groups and branch groups.

COHOMOLOGY Bounded cohomology, L^2 cohomology, and their relation to other subjects, in particular operator algebras.

AMENABILITY Asymptotic properties such as amenability and superamenability, Kazhdan property T, growth, and cogrowth.

ANALYSIS Various topics in Analysis related to groups (in particular spectral theory of discrete Laplace operators on graphs and groups).

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