MenuGroups and Dynamics Seminar
Spring 2024
| Date: | February 7, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Wencai Liu, Texas A&M University |
| Title: | An Invitation to Cocycle |
| Abstract: | 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. |
| Date: | February 14, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Wencai Liu, Texas A&M University |
| Title: | Small denominators in quasi-periodic operators |
| Abstract: | 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. |
| Date: | February 21, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Alain Valette, University of Neuchâtel |
| Title: | Maximal Haagerup subgroups in Zn x SL2(Z) |
| Abstract: | 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. |
| Date: | March 6, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Volodymyr Nekrashevych, Texas A&M University |
| Title: | Conformal dimension and combinatorial modulus |
| Abstract: | 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. |
| Date: | March 20, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Patricia Alonso Ruiz, Texas A&M University |
| Title: | Who is the spectrum of the Sierpinski Gasket? Introductions by an analyst. |
| Abstract: | 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. |
| Date: | March 27, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Jorge Fariña Asategui, Lund University, Sweden |
| Title: | On the Hausdorff dimension of self-similar and branch profinite groups |
| Abstract: | 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. |
| Date: | April 3, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Yuri Bahturin, Memorial University of Newfoundland |
| Title: | Growth of subideals in free Lie algebras and subnormal subgroups in free groups |
| Abstract: | This talk is based on two papers of A. Olshanski (a paper about Lie algebras is joint with the speaker). The papers are devoted to the thorough study of the growth and cogrowth functions of subnormal (subideal) closures of finite sets of elements in the free groups (free Lie algebras). It is interesting that the study of the growth functions often allows one to give answers to the questions in purely abstract form. For example: Any nonzero finitely generated subalgebra of a nonabelian free Lie algebra is self-idealizing, that is, equal to its idealizer. Or: No proper finitely generated subalgebra K of a free Lie algebra L can contain a nonzero subideal H of L. Or else: Let H be a normal subgroup in a free group F with infinite factor group F/H and S a finite subset of H. Then the normal closure N of S in H contains no nontrivial normal subgroups of F. In the group-theoretical paper, the author often refers to the work of R. Grigorchuk. |
| Date: | April 10, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Tatiana Nagnibeda, University of Geneva |
| Title: | On maximal and weakly maximal subgroups in finitely generated groups |
| Abstract: | 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. |
| Date: | April 17, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Dmytro Savchuk, University of South Florida |
| Title: | Explicit Generators for the Stabilizers of Rational Points in Thompson's Group F |
| Abstract: | We construct explicit finite generating sets for the stabilizers in Thompson's group F of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in F. It is well known that the stabilizers of dyadic rational points are isomorphic to F × F and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point b ∈ (0,1) is generated by 5 elements that are explicitly calculated as words in generators x0 , x1 of F that depend on the binary expansion of b. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented. This is a joint work with Krystofer Baker. |
| Date: | April 24, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 123 |
| Speaker: | Santiago Radi, Texas A&M University |
| Title: | On Haar measure of the set of torsion elements in branch pro-p groups |
| Abstract: | In a joint result with J. Fariña-Asategui, we proved that if G is a pro-p branch group, then the Haar measure of the set of torsion elements is zero. |
