Polish groups and Continuous logic – Grupoloco

Our techniques will be based in particular on generalizations of classical model-theoretic notions to the context of metric structures.

Some results obtained by the project's participants and their collaborators:

A correspondence between model-theoretic stability and weakly almost periodic functions in Roelcke-precompact Polish groups. Application: any continuous homomorphism from an omega-categorical structure to a Polish group is open on its range. Equivalently, these groups are totally minimal.

Any Roelcke-precompact non-archimedean Polish group has property (T)

Automorphism groups of countable homegenous (simple, loopless) graphs), direct or not, admit a metrizable universal minimal flow with a comeager orbit.

Any Polish group whose universal minimal flow is metrizable with a generic orbit admits a closed co-precompact extremely amenable subgroup. For these groups, the universal proximal minimal flow may be described explicitly.

Development of a «continuous« version of descriptive set theory; use of this formalism to prove a topometric version of Effros' theorem about comeager orbits for continuous actions of Polish groups.

Full groups of minimal homeomorphisms of a Cantor space do not admit a compatible Polish group topology, and are coanalytic non Borel inside the ambient homeomorphism group. Beginning of a study of closures of such full groups, including a criterion for the existence of a dense conjugacy class.

Members of the project are developing a new and unique blend of techniques from model theory, descriptive set theory, topological dynamics and combinatorics

Members of the project:

- intend to continue working towards a proof (or a counterexample?) of the conjecture according to which any Roelcke precompact Polish group admits a metrizable universal minimal flow with a generic orbit.

- are working on a classification of unitary representations of some «large« Polish groups.

- are analyzing conditions for (non)existence of transitive actions on complete metric spaces of some Polish groups; this question is tied to some fundamental problems about the extension of results and ideas valid for nonarchimedean Polish groups to general Polish groups.

Published or accepted for publication:

1. Itaï Ben Yaacov, The linear isometry group of the Gurarij space is universal, Proceedings of the American Mathematical Society, 142 (2014), no. 7, 2459-2467
2. Itaï Ben Yaacov, Lipschitz functions on topometric spaces, J. Log. Anal. 5 (2013)
3. D. Bilge J. Melleray, Elements of finite order in automorphism groups of homogeneous structures, Contributions to Discrete Mathematics 8, no2 (2013).
4. T. Ibarlucia et J. Melleray, Full group of minimal homeomorphisms and Baire category methods, à paraître à Ergodic Theory and Dynamical Systems .
5. J. Melleray, Extensions of generic measure-preserving actions, à paraître aux Annales de l'Institut Joseph Fourier.
6. L. Nguyen van Thé, More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fund. Math., 222, 19-47, 2013.
7. L. Nguyen van Thé, Universal flows of closed subgroups of S_8, Asymptotic Geometric Analysis, Fields Institute Communications, vol. 68, Springer, 229-245, 2013
8. L. Nguyen van Thé, A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind, 2013, à paraître aux CR Acad. Sci. Serbe.

Submitted for publication:

1. I. Ben Yaacov et A. Kaïchouh, Reconstruction of separably categorical metric structures.
2. I . Ben Yaacov et J. Melleray, Grey subsets of Polish spaces.
3. I. Ben Yaacov et J. Melleray, Isometrisable group actions.
4. I. Ben Yaacov et T. Tsankov, Weakly almost periodic functions, moel-theoretic stability, and minimality of topological groups.
5. D. Evans et T. Tsankov, Free actions of free groups on countable structures and property (T) .
6. J. Jasinski, C. Laflamme, L. Nguyen Van Thé et R. Woodrow, Ramsey precompact expansions of homogeneous directed graphs.
7. J. Melleray, T. Tsankov et L. Nguyen Van Thé, Polish groups with metrizable universal minimal flows.

Our proposal is mainly concerned with the interactions of descriptive set theory and continuous logic, and applications of the methods of these domains to studying various mathematical structures, in particular « large » Polish groups.
Polish groups are the largest well-behaved class of topological groups for which methods of descriptive set theory can be applied. Groups of symmetries of separable objects are usually Polish; prominent examples that are central for our project are the infinite symmetric group S(IN) and closed subgroups of it, the unitary group U(l_2) of the separable infinite-dimensional Hilbert space, the automorphism group Aut(L) of the standard non-atomic probability space, and the isometry group Iso(U) of the Urysohn metric space. Those groups exhibit a variety of interesting behaviours : e.g., automatic continuity of morphisms, extreme amenability, Roelcke precompactness... While many of those properties were initially thought of as pathological, the fact that quite a few « natural » groups satisfy them (in the case, say, of Aut(L), see [GP07] and [BBM*]) or are expected to satisfy them (both U(l_2) and Iso(U) are extremely amenable, cf [GM83] and [Pes02], and automatic continuity is an open question for these groups) has shed a new light on, and made it important to explore, these properties.

There is a long history of interaction between descriptive set theory and model theory, via the study of closed subgroups of S(IN), the permutation group of the integers. Recently, Kechris and Rosendal [KR07] used model-theoretic techniques to classsify subgroups of S(IN) with ample generics. Similarly, Kechris, Pestov and Todorcevic have established an important connection between model theory, combinatorics and topological dynamics in [KPT05], where they characterize extremely amenable subgroups of S(IN) using Fraïssé theory and Ramsey theory.

It has recently come to light that the techniques and concepts of metric model theory, via the recently developed formalism of « continuous logic » (see [BU10] and [BBHU08]) are similarly relevant to the study of Polish groups (see for instance [M10] and [BBM*]). In a related vein, Tsankov has recently used ideas derived from Olshanskii [Ols91] and classical results of model theory to classify the unitary representations of a large class of subgroups of S(IN) , and there is hope that one may use continuous logic to find a similar classification for some large Polish groups.

To put it broadly, the aim of our project is to develop « continuous » versions of techniques and concepts of classical model theory, and use those to gain a better understanding of large Polish groups. Let us give a short list of the type of questions we plan to focus on:

developing of a general theory for automatic continuity and reconstructing group topologies;
classifying the unitary representations of large Polish groups;
developing and applying, a « continuous descriptive set theory », a budding version of which appears in [BM] and has already been applied to obtain some results on Polish groups.

Partial results have already been obtained by the participating members of the project. It seems important to us to push forward and pursue actively research in this direction. To do so, we identify three main needs:
free some time for the project's participants to devote to research on these questions;
develop and popularize this area of research;
visit and invite experts of the areas concerned by the proposal.

We plan to address the first point by asking for funding of sabbaticals for the junior members of the project. For the second point, our plan is to have the proposal's members gather regularly for two-day working seminars and discussion/problem sessions, and organize two concentration weeks, a workshop and an International Conference. Finally, we ask for some travel funds for the proposal's participants and funds to invite leading international experts.