Groups and Dynamics
The purpose of this workshop is to bring together specialists in the theory of groups in dynamics.
Workshop is organized by N. Goncharuk and R. Grigorchuk.
The workshop will follow the colloquium talk by A.Lubotzky, on February 29, 4-5 pm in BLOC 117.
Talks will happen in BLOC 302.
10:00 - 10:50
11:00 - 11:50
11:50 - 1:00
1:00 - 1:50
Aldous-Lyons asked whether every unimodular network is sofic. Equivalently, is every invariant random subgroup of the free group a weak* limit of finite-index IRS’s? I will explain an approach towards proving the answer is `no’. The approach is modeled on the solution to Tsirelson’s problem in the MIP*=RE paper. This is joint work with Michael Chapman, Alex Lubotzky and Thomas Vidick.Back to top
An error-correcting code is locally testable (LTC) if there is a random tester that reads only a small number of bits of a given word and decides whether the word is in the code, or at least close to it. A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance.
Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind. We construct such codes based on what we call (Ramanujan) Left/RightCayley square complexes. These objects seem to be of independent group-theoretic interest. The codes built on them are 2-dimensional versions of the expander codes constructed by Sipser and Spielman (1996).
The main result and lecture will be self-contained. But we hope also to explain how the seminal work of Howard Garland (1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Lie group has led to this construction (even though it is not used at the end). Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes.
Lattices in high-rank semisimple groups enjoy several special properties like super-rigidity, quasi-isometric rigidity, first-order rigidity, and more. In this talk, we will add another one: uniform ( a.k.a. Ulam) stability. Namely, it will be shown that (most) such lattices D satisfy: every finite-dimensional unitary "almost-representation" of D ( almost w.r.t. to a sub-multiplicative norm on the complex matrices) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and Burger-Ozawa-Thom (2013) for SL(n,Z), n>2.
The main technical tool is a new cohomology theory ("asymptotic cohomology") that is related to bounded cohomology in a similar way to the connection of the last one with ordinary cohomology. The vanishing of H^2 w.r.t. to a suitable module implies the above stability.
The talk is based on a joint work with L. Glebsky, N. Monod, and B. Rangarajan.Back to top
We will discuss recent work (with Bridson and Spitler) that constructs examples of finitely presented residually finite groups G that are profinitely rigid amongst finitely presented groups but not amongst finitely generated groups. The key step is to reduce the failure of profinite rigidity to that of Grothendieck pairs, allowing us to construct infinitely many finitely generated subgroups Pn < G for which (G,Pn) is a Grothendieck pair (in particular G and Pn have isomorphic profinite completions).Back to top
Given a group of homeomorphisms of the Cantor set, usually there are many ways to construct a new homeomorphism, as a puzzle, from pieces of several given ones. The group is called ample (or topological full) if it already contains all homeomorphisms obtained that way.
The talk is concerned with the first attempt at a classification of maximal subgroups of ample groups. Results that will be presented are mostly parallel to the classification of maximal subgroups of finite symmetric groups.
Recall that all subgroups of the symmetric group are divided into three classes: intransitive subgroups (those that leave invariant a nontrivial subset), imprimitive subgroups (transitive subgroups that leave invariant a nontrivial partition), and primitive subgroups (the remaining ones). It turns out that the maximal intransitive subgroups are stabilizers of certain subsets while the maximal imprimitive subgroups are stabilizers of certain partitions.
In the case of the ample groups, arbitrary subsets and partitions are replaced by closed subsets and partitions into closed subsets. Transitivity is replaced by minimality (absence of nontrivial closed invariant subsets).
This is joint work with Rostislav Grigorchuk.Back to top