| Date: | May 26, 2020 |
| Time: | Noon |
| Location: | BLOC 164 |
| Speaker: | Mikhail Lyubich, SUNY Stony Brook |
| Title: | Renormalization and spectrum of the Basilica Group. |
| Abstract: | The Basilica Group is the iterated monodromy group associated with the Basilica map z^2-1. The Schur renormalization transformation describes the relation between the Laplacian spectra in various scales. For this group, it happens to be a rational map in two variables. We will describe the dynamical structure of this map that yields the laminar structure for the associated Green current. It implies that the spectrum of this group is a Cantor set of positive Lebesgue measure. It is a work in progress, joint with E.Bedford, N.-B. Dang and R. Grigirchuk.
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| Date: | May 26, 2020 |
| Time: | Noon |
| Location: | BLOC 164 |
| Speaker: | Mikhail Lyubich, SUNY Stony Brook |
| Title: | Renormalization and spectrum of the Basilica Group. |
| Abstract: | The Basilica Group is the iterated monodromy group associated with the Basilica map z^2-1. The Schur renormalization transformation describes the relation between the Laplacian spectra in various scales. For this group, it happens to be a rational map in two variables. We will describe the dynamical structure of this map that yields the laminar structure for the associated Green current. It implies that the spectrum of this group is a Cantor set of positive Lebesgue measure. It is a work in progress, joint with E.Bedford, N.-B. Dang and R. Grigirchuk.
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| Date: | June 3, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Andrew Marks, UCLA |
| Title: | Measurable realizations of abstract systems of congruence |
| Abstract: | In the last few years, several new results have been proved
with the unifying theme that the "paradoxical" sets in many classical
geometrical paradoxes can surprisingly be much "nicer" than one would
naively expect. For example, by the Banach-Tarski paradox any two
bounded subsets A and B of R^3 with nonempty interior are
equidecomposable. However, if A and B have the same Lebesgue measure,
then a recent theorem of Grabowski, M\'ath\'e and Pikhurko states that
A and B are equidecomposable using Lebesgue measurable pieces. So for
example, there is a Lebesgue measurable equidecomposition of a cube
and a ball in R^3 of the same volume
An abstract system of congruences describes a way of partitioning a
space into finitely many pieces satisfying certain congruence
relations. Examples of abstract systems of congruences include
paradoxical decompositions and n-divisibility of actions. We consider
the general question of when there are realizations of abstract
systems of congruences satisfying various measurability constraints.
We completely characterize which abstract systems of congruences can
be realized by nonmeager Baire measurable pieces of the sphere under
the action of rotations on the 2-sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of
congruences for the action of PSL_2(Z) on P^1(R). This is joint work
with Clinton Conley and Spencer Unger. |
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| Date: | June 10, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Mark Sapir, Vanderbilt U. |
| Title: | Groups with quadratic Dehn function and undecidable conjugacy problem |
| Abstract: | This is a joint work with A. Yu. Olshanskii. We construct a finitely
presented group with quadratic Dehn function and undecidable conjugacy
problem. This solves a problem of E. Rips. We also prove that our
group has decidable power conjugacy problem. So it is the first
example of a group with decidable word and power conjugacy problems
and undecidable conjugacy problem. |
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| Date: | June 17, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Bogdan Stankov, ENS, Paris |
| Title: | Non-triviality of the Poisson boundary of certain random walks with finite first moment |
| Abstract: | One equivalent characterization of amenability is the existence of a non-degenerate measure with trivial Poisson boundary (Furstenberg, Rosenblatt, Kaimanovich-Vershik). Vadim Kaimanovich has shown that the Poisson boundary of random walks of finitely supported strictly non-degenerate measures on Thompson's group $F$ is not trivial. It is still not known if $F$ is amenable, which is a famous open question. He asked whether the same statement is true for measures with finite first moment. In this talk we answer that in the positive. More generally, we give a criterion for the non-triviality of the Poisson boundary of random walks on subgroups of groups of piecewise projective homeomorphisms. Furthermore, the simple random walk on the Schreier graph of $F$ has been studied by Mishchenko, who gives another proof of boundary non-triviality. We will show a criterion for non-triviality of an induced random walk on a Schreier graph.
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| Date: | June 24, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Nicolás Matte Bon, ETH Zürich |
| Title: | A commutator lemma for confined subgroups and URS's and application to groups acting on rooted trees. |
| Abstract: | I will explain a commutator lemma for confined subgroups and URSs of a group of homeomorphism, which generalizes to this setting a well-known lemma that relates normal subgroups of a group of homeomorphisms to the rigid stabilizers of the open sets of its action. Then we will focus on various application to weakly branch groups acting on rooted trees.
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| Date: | July 8, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Rachel Skipper, OSU |
| Title: | Generating lamplighter groups with bireversible automata |
| Abstract: | We use the language of formal power series to construct finite state automata generating groups of the form A \wr Z, where A is the additive group of a finite commutative ring and Z is the integers. We then provide conditions on the units of the ring and the power series which make automata bireversible.
This is a joint work with Benjamin Steinberg. |
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| Date: | July 15, 2020 |
| Time: | Noon |
| Location: | 940 9667 3668 |
| Speaker: | Srivatsav Kunnawalkam Elayavalli , Vanderbilt University |
| Title: | Genericity in spaces of enumerated groups |
| Abstract: | We introduce natural Polish space topologies associated the set of enumerated amenable groups G_am, and enumerated small groups G_sm (groups that don't admit an embedding of F_2, the free group on 2 generators), and then show that the nonamenable small groups are co-meager in G_sm, and non elementary amenable amenable groups are co-meager in G_am, thereby providing a generic solution to the von Neumann-Day problem. In connection with these ideas, we will discuss additional topics such as model theoretic forcing, theories of amenable groups, and some interesting open questions that arise from this work. This is based on recent joint work with I. Goldbring.
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| Date: | July 22, 2020 |
| Time: | Noon |
| Location: | ID: 940 9667 366 |
| Speaker: | Peter Kuchment, Texas A&M |
| Title: | Extensions of group representations and functor Ext^1 in the category of Banach spaces. |
| Abstract: | Let one have a representation T of a group G in a HIlbert space H and E be a closed invariant subspace. Then T generates a sub-representation T_1 in E, as well as quotient-representation T_2 in the quotient space F:=H/E. What additional information, besides T_1 and T_2 is needed to recover the presentation T? The answer in the finite-dimensional or in HIlbert spaces case is well known: one needs a group cohomology class h in H^1_G(L(F,E)), where L(F,E) is the space of bounded linear operators from F to E. This result hinges upon existence of a complement to the space E in H. However, it is known that in any Banach space that is not isomorphic to Hilbert one, there exist non-complemented subspaces. This makes the problem (formulated 50 years ago by A. Kirillov in his group representation textbook) non-trivial even for trivial group actions. The speaker announced without proof 45 years ago a solution of this problem. It also entailed studying the functor Ext^1 in the category of Banach spaces. The author's work on this, besides the Kirillov's problem, was triggered by the wonderful paper by Enflo, Lindenstrauss, and Pisier on extensions of Hilbert spaces. A few years later, starting with the work by N. Kalton, such a study, named "the three-space problem" started flourishing and has been developed significantly since. However, approaches to and results on Ext^1 of the speaker's paper mostly have not been rediscovered, to the best of the author's knowledge. The talk will contain the results and sketches of the proofs. (A preprint is available on arXiv.) |
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