| Date: | January 31, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Rostislav Grigorchuk, Texas A&M |
| Title: | Group of intermediate growth, aperiodic order, and Schroedinger operators. |
| Abstract: | I will explain how seemingly unrelated objects: the group G of intermediate growth constructed by the speaker in 1980, the aperioidc order, and the theory of (random) Schroediinger operator can meet together. The main result, to be discussed, is based on a joint work with D.Lenz and T.Nagnibeda. It show that a random Markov operator on a family of Schreier graphs of G associated with the action on a boundary of a binary rooted tree has a Cantor spectrum of the Lebesgue measure zero. This will be used to gain some information about the spectrum of the Cayley graph. The main tool of investigation is given by a substitution, that, on the one hand, gives a presentation of G in terms of generators and relations, and, on the other hand, defines a minimal substitutional dynamical system which leads to the use of the theory of random Shroedinger operator.
No special knowledge is assumed, and the talk is supposed to be easily accessible for the audience.
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| Date: | February 7, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Volodymyr Nekrashevych, Texas A&M University |
| Title: | Examples of simple groups of intermediate growth |
| Abstract: | I will describe several explicit examples of simple groups of intermediate growth: groups associated with irrational rotations, a group acting on the Morse-Thue subshift, an embedding of the Grigorchuk group into a simple group of intermediate growth, etc.. Open questions and directions for future research will be also discussed. |
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| Date: | February 14, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Bernhard Reinke, Jacobs University, Bremen |
| Title: | Iterated Monodromgy Groups of entire functions |
| Abstract: | Iterated Monodromy Groups can also be defined for post-singularly finite transcendental functions. They have a self-similar action on a regular rooted tree, but in contrast to IMGs of rational functions, every vertex of the tree has infinite degree. In my talk, I will focus mainly on the exponential family, where we can show that the associated IMG is amenable, using an explicit description of the IMG in terms of the kneading sequence of the exponential map. |
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| Date: | February 28, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Yaroslav Vorobets, Texas A&M |
| Title: | Finite generation of topological full groups |
| Abstract: | Topological full groups provide a connection between topological dynamics and group theory. A countable transformation group is associated to a dynamical system and, as an abstract group, it completely encodes the dynamics.
It is known that the derived group of the topological full group is finitely generated provided that dynamics is minimal and expansive. However the group itself need not be finitely generated. I am going to discuss issues with finite generation and then present a new example of a finitely generated topological full group.
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| Date: | March 1, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Robin Tucker-Drob, Texas A&M |
| Title: | Superrigidity and Measure Equivalence |
| Abstract: | Measure Equivalence is an equivalence relation on countable groups introduced by Gromov as a measure-theoretic counterpart to the relation of quasi-isometry coming from geometric group theory. In the first part of this talk I will give a brief introduction to measure equivalence, focusing on various known invariants of measure equivalence.
In the second part of the talk I will discuss some recent work in which I show that the collection of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. I will not say too much about the proof, but rather i plan to discuss some consequences of this result as well as some related open questions. |
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| Date: | April 4, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Dzmitry Dudko, Stony Brook University |
| Title: | Conjugacy problem in mapping class bisets |
| Abstract: | For a branched covering f of a sphere, the mapping class biset M(f) is the set of maps obtained by pre- and post-composing f with elements of the mapping class group. Maps in M(f) are considered up to isotopy relative the postcritical set. We will show that the conjugacy problem in M(f), known also as Thurston equivalence, is decidable. |
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| Date: | April 18, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Guoliang Yu, Texas A&M |
| Title: | A friendly introduction to the Baum-Connes conjecture |
| Abstract: | In this talk, I will give an introduction to the Baum-Connes conjecture and discuss its applications. I will explain how geometry of groups comes into the picture. |
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| Date: | April 25, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Guoliang Yu, Texas A&M |
| Title: | A friendly introduction to the Baum-Connes conjecture (part 2) |
| Abstract: | In this talk, I will give an introduction to the Baum-Connes conjecture and discuss its applications. I will explain how geometry of groups comes into the picture. |
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| Date: | May 2, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Dmytro Savchuk, University of South Florida |
| Title: | Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers |
| Abstract: | We show that every polynomial in $\mathbb Z[x]$ defines an endomorphism of the $d$-ary rooted tree induced by its action on the ring $\mathbb Z_d$ of $d$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $\mathbb Z[x]$ of the same degree. In the case of permutational polynomials acting on $\mathbb Z_d$ by bijections the induced endomorphisms are automorphisms of the tree. In the case of $\mathbb Z_2$ such polynomials were completely characterized by Rivest. As our main application we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial $f(x)\in\mathbb Z[x]$ that is necessary and sufficient for $f$ to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of $f(x)$ on $\mathbb Z_2$ with respect to the normalized Haar measure. |
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