| Date: | August 29, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Bruno Duchesne, Élie Cartan Institute of Lorraine |
| Title: | Groups acting on dendrites |
| Abstract: | A dendrite is a compact arcwise-connected locally connected metrizable space such that any two points are joined by a unique arc. One can think to them as generalizations of compactified trees.
We will be interested in groups acting on dendrites by homeomorphisms and examples of such groups. In particular, I will explain the construction of kaleidoscopic groups than can be thought as analogs of Burger-Mozes universal groups for dendrites. This is a joint work with Nicolas Monod and Phillip Wesolek. |
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| Date: | September 5, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Anush Tserunyan, UIUC |
| Title: | Ergodic hyperfinite decomposition of pmp equivalence relations |
| Abstract: | A countable Borel equivalence relation $E$ on a probability space can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group and as the connectedness relation of a locally countable Borel graph, called a \emph{graphing} of $E$. When $E$ is probability measure preserving (pmp), graphings provide a numerical invariant called \emph{cost}, whose theory has been largely developed and used by Gaboriau and others in establishing rigidity results. A well-known theorem of Hjorth states that when $E$ is pmp, ergodic, treeable (admits an acyclic graphing), and has cost $n \in \mathbb{N} \cup \{\infty\}$, then it is generated by an a.e. free measure-preserving action of the free group $\mathbf{F}_n$ on $n$ generators. In our work with Benjamin Miller, we develop techniques of modifying the graphing, which yield a strengthening of this theorem: the action of $\mathbf{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically. |
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| Date: | October 10, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Slava Grigorchuk |
| Title: | On the question: "Can one hear the shape of a group?" and Hulancki type theorem for graphs |
| Abstract: | In my talk I will address the famous question of M.Kac (traced back to L.Bers and A.Weyl) "Can one hear the shape of a drum?" in the context of groups viewed as geometric objects. I will show that the answer in NO in a strong sense: there is a continuum family of 4-generated pairwise not quasi-isometric groups with the same spectrum of discrete Laplacian. Moreover each of these groups has uncountable family of amenable covering groups with the same spectrum.
The arguments will be based on the construction by the speaker of groups of intermediate growth (between polynomial and exponential), and the results in the spectral theory of graphs which somehow is related to the famous Hulanicki criterion of amenability of groups in terms of weak containment of unitary representations. The talk is based on joint results with A.Dudko. |
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| Date: | October 31, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Constantine Medynets, US Naval Academy |
| Title: | Continuous Orbit Equivalence Rigidity |
| Abstract: | In the 1980s, Mike Boyle proved that whenever two minimal
Z-actions on the Cantor set are continuous orbit equivalent, they are
automatically conjugate. In the talk, we will discuss the phenomenon of
continuous orbit equivalence rigidity and what systems are known to
exhibit it. |
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| Date: | November 14, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Anton Bernshteyn, CMU |
| Title: | Is multiplication of weak equivalence classes continuous? |
| Abstract: | The relations of weak containment and weak equivalence were introduced by Kechris in order to provide a convenient framework for describing global properties of p.m.p. actions of countable groups. Weak equivalence is a rather coarse relation, which makes it relatively well-behaved; in particular, the set of all weak equivalence classes of p.m.p. actions of a given countable group $\Gamma$ carries a natural compact metrizable topology. Nevertheless, a lot of useful information about an action (such as its cost, type, etc.) can be recovered from its weak equivalence class. In addition to the topology, the space of weak equivalence classes is equipped with a multiplication operation, induced by taking products of actions, and it is natural to wonder whether this multiplication operation is continuous. The answer is positive for amenable groups, as was shown by Burton, Kechris, and Tamuz. In this talk, we will explore what happens in the nonamenable case. Number theory will make an appearance.
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| Date: | November 28, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Roman Kogan |
| Title: | Graphs of Action and the Automatic Logarithm |
| Abstract: | We introduce a new construction, called the Automatic
Logarithm, motivated by the study of graphs of action of the group
generated by two Mealy automata, A and B, on the levels of the infinite
rooted binary tree T, in the case where A is level-transitive and of
bounded activity. The automatic logarithm L_{A,B} computes the length of
chords in these graphs. As a function form the boundary of the tree dT
to dyadics Z_2, its values are given by a Moore machine whose output is
interpreted as a dyadic integer. |
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| Date: | November 30, 2018 |
| Time: | 2:30pm |
| Location: | BLOC 220 |
| Speaker: | Zoran Sunic |
| Title: | Hanoi Towers Groups on 3 or more pegs |
| Abstract: | The group H is a finitely generated, self-similar group acting on a rooted ternary tree in such a way that the Schreier graph of the action on level N models the Hanoi Towers game played with N disks (the vertices represent the possible configurations and the group generators the possible moves).
The group H has many interesting properties in its own right:
- It is amenable but not subexponentially amenable group.
- Calculations involving finite dimensional permutational representations of H based on the self-similarity of the group lead to calculation of the spectra of the Sierpiński graphs.
- It is the iterated monodromy group of a post-critically finite rational map on the Riemann sphere.
- Its closure is finitely constrained (in the sense of symbolic dynamics on trees).
- Its Hausdorff dimension is irrational and its limit space is the well known Sierpiński gasket.
- It is the first first known example of a finitely generated branch group that maps onto the infinite dihedral group.
- It was the first known example of a finitely generated branch group with nontrivial rigid kernel.
- It has exponential growth.
- It contains a copy of every finite 3-group.
- ...
There are also groups associated with versions of the game on more than 3 pegs. These “higher Hanoi Towers groups” are perhaps even more interesting, but also more difficult to study. For instance, it is known that they do not contain free subgroups, but it is not known if they are amenable. |
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