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Group theory seminar Yassawi/Tsankov/Kellendonk

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Group theory seminar Yassawi/Tsankov/Kellendonk

December  13 (wednesday, unusual day)

14.00–14.45 Reem Yassawi (Queen Mary University of London) « Tame or wild Toeplitz shifts »

15.00–15.45 Todor Tsankov (Lyon 1) « Gleason complete flows of locally compact groups »

16.15–17.00 Johannes Kellendonk (Lyon 1) « Which algebraic components of the Ellis semigroup of a non-tame dynamical system are especially big? »

Reem Yassawi     « Tame or wild Toeplitz shifts« 

The Ellis semigroup E(X, T) of a topological dynamical system is defined to be the compactification of the action T in the topology of pointwise convergence on the space of all functions X^X. Tameness is a concept whose roots date back to Rosenthal’s ℓ^1 embedding theorem, which says that if a sequence in ℓ^1 does not have a weakly Cauchy subsequence, then it must be the sequence of unit vectors in ℓ^1. Köhler linked the concept of tameness to the Ellis semigroup. A system is tame if its Ellis semigroup has size at most the continuum. Non-tame systems are very far from tame, as they must contain a copy of βℕ, the Stone-Čech compactification of ℕ.

Since then, the dynamics community has investigated the question of which systems are tame. In this talk I will give a brief exposition of these results, and talk about work where we study tameness, or otherwise, of Toeplitz shifts, emphasizing the connection between this work and automata. This is joint work with Gabriel Fuhrmann and Johannes Kellendonk.

Todor Tsankov « Gleason complete flows of locally compact groups« 

The notion of an irreducible extension of a flow generalizes the one of an almost one-to-one extension (injective on a dense G_delta set)and coincides with the one of a highly proximal extension for minimal flows. The existence of maximal such extensions was proved byAuslander and Glasner in the 70s for minimal flows using an abstract argument, and a concrete construction using near-ultrafilters wasrecently given by Zucker for arbitrary flows. When the acting group is discrete, the universal irreducible extension is nothing but the Stonespace of the Boolean algebra of the regular open sets of the space, already considered by Gleason. We give yet another construction of theuniversal irreducible extension for arbitrary topological groups and prove that for such extensions (which we call Gleason complete) ofa flow of a locally compact group G, the stabilizer map x -> G_x is continuous (for general flows, this map is only semi-continuous). Thisis a common generalization of a theorem of Frolík that the set of fixed points of a homeomorphism of a compact, extremally disconnectedspace is open and a theorem of Veech that the action of a locally compact group on its greatest ambit is free. The theorem implies, in particular, that if the action of a locally compact group on its Furstenberg boundary is essentially free, then it is free. This is joint work with Adrien Le Boudec.

Johannes Kellendonk « Which algebraic components of the Ellis semigroup of a non-tame dynamical system are especially big?« 

The Ellis semigroup E of a group acting by homeomorphisms on a compact space is its compactification in the topology of point wise convergence. It has a lot of interesting structures:  its topology, the topological properties of its elements, and its algebraic structure. One property which has incited of lot of interest in recent years is tameness.  In can be characterised in various different ways, but for our talk the quickest way is to say is that E is tame if its cardinality is at most that of the continuum. So non-tame Ellis semigroups are especially big. We are interested in how this relates to the algebraic structure of the Ellis semigroup. For instance, when is the kernel of the Ellis semigroup especially big? A recent result shows that, if the set of idempotents of a minimal right ideal of the Ellis-semigroup of a minimal system is especially big, then the system cannot be a PI-flow. It is also known that for minimal actions of groups which do not carry an invariant measure, tameness implies that the system is almost automorphic. In both cases the converse is not true (for almost automorphic non-tame systems see in particular the talk by Reem Yassawi). We will show here that for minimal abelian group actions which are not almost automorphic and whose set of singular points satisfies a condition which will be specified, the kernel of the Ellis semigroup is especially big, and here it is in particular the Rees structure group which is especially big.

Organized by Andrei Alpeev, Laurent Bartholdi, Anna Erschler and Panagiotis Tselekidis 

Partially supported by ERC Advanced Grant 101097307 (P.I.:Laurent Bartholdi).

- Séminaire de théorie des groupes à l’ENS

Détails :

Orateur / Oratrice : Reem Yassawi, Todor Tsankov, Johannes Kellendonk
Date : 13 décembre 2023
Horaire : 14h00 - 17h00
Lieu : 14:00-17:00 Salle W