Groups and Dynamics Seminar
Organizers:
Rostislav Grigorchuk,
Volodia Nekrashevych,
Zoran Šunić, and
Robin Tucker-Drob.
Arman Darbinyan
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Date Time |
Location | Speaker |
Title – click for abstract |
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02/07 3:00pm |
BLOC 123 |
Wencai Liu Texas A&M University |
An Invitation to Cocycle
The cocycle is fundamentally important in the analysis of Schrodinger operators, serving as a bridge between dynamical systems tools and spectral theory. In this talk, I will introduce a variety of techniques for investigating both quasiperiodic and random Schrodinger operators. Topics covered will include the regularity of Lyapunov exponents and rotation numbers, the topological structure of the spectrum, and the non-uniform hyperbolicity of cocycles. If time permits, I will also discuss some open problems in the field. |
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02/14 3:00pm |
BLOC 123 |
Wencai Liu Texas A&M University |
Small denominators in quasi-periodic operators
This presentation will explore two approaches for tackling the small denominators problem in the analysis of quasi-periodic Schrödinger operators. We begin by applying a KAM (Kolmogorov-Arnold-Moser) type method to establish irreducibility. Following this, we discuss an alternative approach that leverages semi-algebraic geometry and Cartan-type estimates. |
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02/21 3:00pm |
BLOC 123 |
Alain Valette University of Neuchâtel |
Maximal Haagerup subgroups in Zn x SL2(Z)
The Haagerup property is a strong negation of Kazhdan's property (T). In a countable group, every Haagerup subgroup is contained in a maximal one. We propose to classify maximal Haagerup subgroups in the semi-direct product Gn=Zn x SL2(Z), where the action of SL2(Z) on Zn is induced by the unique irreducible representation of SL2(R) on Rn (with n>1). We prove that there is a dichotomy for maximal Haagerup subgroups in Gn: either (amenable case) they are of the form Zn x K, with K maximal amenable in SL2(Z); or (non-amenable case) they are transverse to Zn. This extends work by Jiang and Skalski for n=2. In joint work with P. Jolissaint, for n even, we prove the stronger result that the von Neumann algebra of Zn x K (K as above) is maximal Haagerup in the von Neumann algebra of Gn. This involves looking at the orbit equivalence relation induced by SL2(Z) on the n-torus, and proving that it satisfies a dichotomy: every ergodic sub-equivalence relation is either rigid or hyperfinite. This extends a result by Ioana for n=2. |
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03/06 3:00pm |
BLOC 123 |
Volodymyr Nekrashevych Texas A&M University |
Conformal dimension and combinatorial modulus
We will discuss the Ahlfors-regular conformal dimension of the limit space of a self-similar group. In particular, I will give a characterization of the conformal dimension in terms of the combinatorial modulus of the dual Moore diagram of the automaton generating the group. This is an adaptation of the characterization of the conformal dimension due to Carrasco Piaggio and Keith-Kleiner. We will also discuss a characterization of the dimension in terms of the Schatten class of commutators of natural representations of the group and the algebra of continuous functions on the limit space.
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03/20 3:00pm |
BLOC 123 |
Patricia Alonso Ruiz Texas A&M University |
Who is the spectrum of the Sierpinski Gasket? Introductions by an analyst.
At a party over spring break, a group theorist and an analyst colleague discover they are both acquainted with a fractal called the Sierpinski gasket.
"Are you also familiar with its spectrum?'' the group theorist asks.
"Indeed!'' the analyst replies with excitement.
"It is remarkably interesting: a rather precise characterization, exponentially growing gaps, an explicit minimal gap, and still some features remain unknown...''.
The two colleagues decide to find a whiteboard to formally portray the spectrum as they have got to known it. This talk will present the analyst's (still incomplete) picture.
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03/27 3:00pm |
BLOC 123 |
Jorge Fariña Asategui Lund University, Sweden |
On the Hausdorff dimension of self-similar and branch profinite groups
Groups acting on regular rooted trees provide easy examples of groups with exotic properties such as Burnside groups and groups of intermediate growth. Of particular interest are branch profinite groups as they constitute one of the two classes partitioning the class of just infinite profinite groups. Based on the work of Abercrombie, Barnea and Shalev started the study of the Hausdorff dimension on profinite groups. The Hausdorff dimension of self-similar profinite groups is still the object of several open problems. The first part of this talk is devoted to introducing groups acting on regular rooted trees and the Hausdorff dimension of their closures. Then we introduce a new tool to compute the Hausdorff dimension of the closure of a self-similar group. Using this new tool we solve an open problem of Grigorchuk on the self-similar Hausdorff spectrum of the group of q-adic automorphisms. Indeed, we completely determine the Hausdorff spectra of the group of q-adic automorphisms restricted to different classes of closed subgroups. We also solve a well-known open problem of Boston on the Hausdorff dimension of just infinite branch pro-p groups. Lastly, if time permits, we will discuss some open problems on the finitely generated Hausdorff spectrum of branch profinite groups and some new results in this direction on an ongoing joint project with Garaialde Ocaña and Uria-Albizuri. |
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04/10 3:00pm |
BLOC 123 |
Tatiana Nagnibeda University of Geneva |
On maximal and weakly maximal subgroups in finitely generated groups
Margulis and Soifer proved that in a finitely generated linear group all maximal subgroups are of finite index if and only if the group is solvable; otherwise there exist uncountably many maximal subgroups of infinite index, all of them isomorphic to a free group of infinite rank. In the talk we will discuss maximal and weakly maximal subgroups in some non-linear finitely generated groups, such as branch and weakly branch groups and Thompson’s group F. |
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).