# Structure and Geometry of Polish groups (17w5094)

Arriving in Oaxaca, Mexico Sunday, June 11 and departing Friday June 16, 2017

## Organizers

Christian Rosendal (University of Illinois at Chicago)

Julien Melleray (Université Lyon 1)

Pierre-Emmanuel Caprace (Université catholique de Louvain)

Stevo Todorcevic (University of Toronto)

## Objectives

(1) Scientific program: The principal aim of the proposed workshop is to bring together people from different areas of mathematics for whom Polish groups intervene in an essential manner in their research or for whom these may even be the main object of study.

To fix ideas, a Polish group is a separable topological group whose topology may be induced by a complete metric. These appear naturally all over mathematics as topological transformation groups, e.g., $$ {\rm Homeo}(\mathcal M), \;\; {\rm Diff}(\mathcal M),\;\; {\rm Isom}(X,\|\cdot\|),\;\; {\rm Aut}(\bf A), $$ where $\mathcal M$ is a compact (smooth) manifold, $X$ is a separable Banach space and $\bf A$ is some countable discrete structre. Given their ubiquity, it is not surprising that these have received substantial attention. Indeed, various subclasses such as Banach spaces, locally compact groups, infinite-dimensional Lie groups, homeomorphism and diffeomorphism groups have been intensely studied from various perspectives, but only recently, essentially in the last 20 years, have they come together as a unified class of objects to be studied. Part of the impetus for this came from logic, in particular, descriptive set theory, where the exact condition of being Polish is essential in the work of classifying the complexities of orbital equivalence relations induced by continuous group actions, which is related to the determination of complexities of classification problems. Though this motivation is still relevant, much of recent work on Polish groups within logic has focused on other aspects and the subject has similarly flourished elsewhere. In the following, we will describe a few of the many topics that are currently under investigation.

(2) Topological dynamics, ergodic theory and harmonic analysis: A program in ergodic theory long advocated by G. Mackey and B. Weiss has been to investigate ergodic theoretical concepts under the more precise language of descriptive set theory, meaning without discarding sets of measure zero. As an example, consider an action $G\curvearrowright ([0,1],\lambda)$ of a group $G$ by measure-preserving transformations, i.e., so that each $g\in G$ defines, up to measure zero, a measure-preserving transformation. One may ask when such an action may be exactly implemented, i.e., implemented by a Borel measurable action $G\curvearrowright [0,1]$. Though this may fail for a general Polish group $G$, Mackey showed that this can always be done provided $G$ is locally compact and recent work by a number of people, including H. Becker, E. Glasner, A. Kwiatkowska, J. Moore, S. Solecki, A. Tornquist, B. Tsirelson and B. Weiss has focused on detailing the maximal extent of Mackey's result.

A particularly fertile line of research in this area has revolved around the concept of extreme amenability. Here a topological group $G$ is extremely amenable if every continuous $G$-action on a compact Hausdorff space has a fixed point. As shown by W. Veech, no non-trivial locally compact group is extremely amenable, but examples of Polish extremely amenable groups were constructed by J.P.R. Christensen and W. Herer. Connections between the concentration of measure phenomenon and the extreme amenability of the unitary group were established by M. Gromov and V. Milman, but the theory really started developing with the work of V. Pestov in the 1990s and the subsequent connection between extreme amenability of automorphism groups of first-order structures and structural Ramsey theory of finite structures developed in an influential paper by A. Kechris, Pestov and S. Todorcevic. The determination of model theoretical classes with structural Ramsey properties and further development of the dynamical aspects of the theory continues to be the focus of research with a number of younger people working in the area, e.g., J. Melleray, L. Nguyen van The, T. Tsankov and A. Zucker.

Other studies with a harmonic analytical flavour have focused on compactifications of Polish groups, in particular, V. Uspenskii has developed a substantial theory of Roelcke precompact groups, i.e., precompact in the Roelcke uniformity. Moreover, as the Roelcke precompact automorphism groups turn out to be essentially the automorphism groups of countably categorical structures, this perspective is particularly fruitful in connection with model theory, where Tsankov has been able to classify unitary representations of such groups and, in collaboration with I. Ben Yaacov, studied weakly almost periodic functions on these. Again this ties up with research by Glasner and M. Megrelishvili on compactifications associated to dynamical systems.

(3) Topological rigidity: A phenomenon surprisingly prevalent in Polish groups is topological rigidity, that is, that the topological structure on the groups is completely determined by the underlying algebraical features of these. In many ways, such matters have been known for many years in the measure-theoretical context, for example, in the form of the automatic continuity of measurable homomorphisms between locally compact groups. Similarly, motivated by results of G. Ahlbrandt and M. Ziegler, many automorphism groups of countable first-order structures have been shown to have an essentially unique permutation group structure.

However, more recently, elaborating on earlier work of D. Lascar, W. Hodges, I. Hodkinson and S. Shelah, a much stronger order of rigidity was shown to hold in groups with large conjugacy classes, specifically with ample generics, by A. Kechris and C. Rosendal. Namely, every homomorphism from such a group into a separable topological group is automatically continuous. The extent of this phenomenon has subsequently been greatly expanded, e.g., to all homeomorphism groups of compact manifolds by K. Mann, Rosendal and Solecki and a host of other naturally occurring transformation groups by Ben Yaacov, A. Berenstein, Melleray, Sabok and others.

The underlying technique of ample generics or simply large conjugacy classes in Polish groups remains of vital importance, both in terms of the structural rigidity mentioned above, but also for the understanding of generic properties of single automorphisms of the underlying phase space of the transformation groups. Again, the limits and extent of this has been studied by, e.g., A. Kaichouh, A. Kwiatkowska, F. LeMaitre, M. Malicki, K. Slutsky and P. Wesolek.

Also, in geometric topology several studies revolving around the question of which homeomorphism or diffeomorphism groups may act on what manifolds have been conducted. Questions of this type appear, for example, in the work of E. Ghys, but are perhaps best motivated by problems of extension. E.g., if $M$ and $N$ are compact manifolds with $M\subseteq N$, when and how does the action of ${\rm Homeo}(M)$ on $M$ extend to an action on $N$ and, if so, is the extension always continuous? Such questions have been studied, e.g., by D. Epstein and V. Markovic and more recently by S. Hurtado, Mann and E. Militon and automatic continuity proves to be a fundamental tool in this context.

(4) Geometric group theory: Geometric group theory or large scale geometry of finitely generated groups stimulated primarily by the work of Gromov is currently an area of high activity and interest, which permeates many areas of mathematics. Over the last 15 years, the large scale geometry of locally compact groups has also come into focus, both in the setting of Lie groups and for totally disconnected groups (a book surveying this is in preparation by Y. de Cornulier and P. de la Harpe). Research in this area connects naturally with the study of harmonic analytic aspects of locally compact groups and $C^*$-algebra techniques, e.g., rigdity and antirigidity properties such as Kazhdan's property (T) and the Haagerup property as evidenced in work by B. Bekka, de la Harpe, Y. Shalom, R. Tessera and A. Valette among many others. Lately, under the impetus of papers by D. Fisher and G. Margulis and by U. Bader, A. Furman, T. Gelander and N. Monod, the study of linear and affine actions on Banach spaces more general than Hilbert space has received a substantial amount of attention leading to an expanded toolset of relevance to the broader class of Polish groups.

Whereas non-locally compact Polish groups do not immediately lend themselves to the techniques of geometric group theory, recent discoveries by Rosendal allow one to transfer a large part of the machinery from the discrete or locally compact setting to all Polish groups without exception. This provides a unified conceptual framework in which geometric group theory and geometric non-linear functional analysis, i.e., the non-linear analysis of Banach spaces, are different instances of a single category and reveals deeper coarse geometric features of general Polish groups.

(5) Other areas: Many other areas have a vested interest in the structure of Polish groups or provide ideas for its further development. Of specific examples, one can mention the general study of infinite closed permutation groups, i.e., closed subgroups of the group all permutations of an infinite set, and its connections with model theory represented by researchers such as P. Cameron, D. Macpherson, P. Neumann, K. Tent and S. Thomas. Similarly, the permutation groups among locally compact groups are exactly the totally disconnected locally compact groups, a topic that has received substantial attention over the last 20 years through seminal work of G. Willis and was recently the subject of an Arbeitsgemeinschaft in Oberwolfach organised by P.-E. Caprace and Monod. And, finally, infinite-dimensional Lie groups, which from the pure side has received considerable attention, e.g., by H. Glockner and K.-H. Neeb.

(6) Workshop program: As evidenced by the flourishing activity in a broad range of areas, Polish groups is a vibrant field. Though partially represented in various conferences organised by different communities, thus far there has been no conference whose principal focus is the structure of Polish groups. Given the maturity of the subject and the unifying threads identified above, we feel that the time is ripe for a conference dedicated to this particular topic.

With the aim of ensuring the interdisciplinarity of the conference and to facilitate collaboration and transfer of knowledge between the different research groups, the intended participants would be selected from many different areas of mathematics in which Polish groups play a central role. Of course, as Polish groups encompass, e.g., all compact Lie groups and separable Banach spaces, some restriction is in order. As criteria for inclusion, we would select researchers whose prime interests are not entirely covered by the structure theory of classical, i.e., finite-dimensional Lie groups or whose concerns are closely related to the problems mentioned above. Also, several other factors have entered in identifying a list of potential participants, namlely, geographical variation and a proper represenation of women scholars. Similarly, we have focused on including younger people, students and post-docs in the line-up.

To fix ideas, a Polish group is a separable topological group whose topology may be induced by a complete metric. These appear naturally all over mathematics as topological transformation groups, e.g., $$ {\rm Homeo}(\mathcal M), \;\; {\rm Diff}(\mathcal M),\;\; {\rm Isom}(X,\|\cdot\|),\;\; {\rm Aut}(\bf A), $$ where $\mathcal M$ is a compact (smooth) manifold, $X$ is a separable Banach space and $\bf A$ is some countable discrete structre. Given their ubiquity, it is not surprising that these have received substantial attention. Indeed, various subclasses such as Banach spaces, locally compact groups, infinite-dimensional Lie groups, homeomorphism and diffeomorphism groups have been intensely studied from various perspectives, but only recently, essentially in the last 20 years, have they come together as a unified class of objects to be studied. Part of the impetus for this came from logic, in particular, descriptive set theory, where the exact condition of being Polish is essential in the work of classifying the complexities of orbital equivalence relations induced by continuous group actions, which is related to the determination of complexities of classification problems. Though this motivation is still relevant, much of recent work on Polish groups within logic has focused on other aspects and the subject has similarly flourished elsewhere. In the following, we will describe a few of the many topics that are currently under investigation.

(2) Topological dynamics, ergodic theory and harmonic analysis: A program in ergodic theory long advocated by G. Mackey and B. Weiss has been to investigate ergodic theoretical concepts under the more precise language of descriptive set theory, meaning without discarding sets of measure zero. As an example, consider an action $G\curvearrowright ([0,1],\lambda)$ of a group $G$ by measure-preserving transformations, i.e., so that each $g\in G$ defines, up to measure zero, a measure-preserving transformation. One may ask when such an action may be exactly implemented, i.e., implemented by a Borel measurable action $G\curvearrowright [0,1]$. Though this may fail for a general Polish group $G$, Mackey showed that this can always be done provided $G$ is locally compact and recent work by a number of people, including H. Becker, E. Glasner, A. Kwiatkowska, J. Moore, S. Solecki, A. Tornquist, B. Tsirelson and B. Weiss has focused on detailing the maximal extent of Mackey's result.

A particularly fertile line of research in this area has revolved around the concept of extreme amenability. Here a topological group $G$ is extremely amenable if every continuous $G$-action on a compact Hausdorff space has a fixed point. As shown by W. Veech, no non-trivial locally compact group is extremely amenable, but examples of Polish extremely amenable groups were constructed by J.P.R. Christensen and W. Herer. Connections between the concentration of measure phenomenon and the extreme amenability of the unitary group were established by M. Gromov and V. Milman, but the theory really started developing with the work of V. Pestov in the 1990s and the subsequent connection between extreme amenability of automorphism groups of first-order structures and structural Ramsey theory of finite structures developed in an influential paper by A. Kechris, Pestov and S. Todorcevic. The determination of model theoretical classes with structural Ramsey properties and further development of the dynamical aspects of the theory continues to be the focus of research with a number of younger people working in the area, e.g., J. Melleray, L. Nguyen van The, T. Tsankov and A. Zucker.

Other studies with a harmonic analytical flavour have focused on compactifications of Polish groups, in particular, V. Uspenskii has developed a substantial theory of Roelcke precompact groups, i.e., precompact in the Roelcke uniformity. Moreover, as the Roelcke precompact automorphism groups turn out to be essentially the automorphism groups of countably categorical structures, this perspective is particularly fruitful in connection with model theory, where Tsankov has been able to classify unitary representations of such groups and, in collaboration with I. Ben Yaacov, studied weakly almost periodic functions on these. Again this ties up with research by Glasner and M. Megrelishvili on compactifications associated to dynamical systems.

(3) Topological rigidity: A phenomenon surprisingly prevalent in Polish groups is topological rigidity, that is, that the topological structure on the groups is completely determined by the underlying algebraical features of these. In many ways, such matters have been known for many years in the measure-theoretical context, for example, in the form of the automatic continuity of measurable homomorphisms between locally compact groups. Similarly, motivated by results of G. Ahlbrandt and M. Ziegler, many automorphism groups of countable first-order structures have been shown to have an essentially unique permutation group structure.

However, more recently, elaborating on earlier work of D. Lascar, W. Hodges, I. Hodkinson and S. Shelah, a much stronger order of rigidity was shown to hold in groups with large conjugacy classes, specifically with ample generics, by A. Kechris and C. Rosendal. Namely, every homomorphism from such a group into a separable topological group is automatically continuous. The extent of this phenomenon has subsequently been greatly expanded, e.g., to all homeomorphism groups of compact manifolds by K. Mann, Rosendal and Solecki and a host of other naturally occurring transformation groups by Ben Yaacov, A. Berenstein, Melleray, Sabok and others.

The underlying technique of ample generics or simply large conjugacy classes in Polish groups remains of vital importance, both in terms of the structural rigidity mentioned above, but also for the understanding of generic properties of single automorphisms of the underlying phase space of the transformation groups. Again, the limits and extent of this has been studied by, e.g., A. Kaichouh, A. Kwiatkowska, F. LeMaitre, M. Malicki, K. Slutsky and P. Wesolek.

Also, in geometric topology several studies revolving around the question of which homeomorphism or diffeomorphism groups may act on what manifolds have been conducted. Questions of this type appear, for example, in the work of E. Ghys, but are perhaps best motivated by problems of extension. E.g., if $M$ and $N$ are compact manifolds with $M\subseteq N$, when and how does the action of ${\rm Homeo}(M)$ on $M$ extend to an action on $N$ and, if so, is the extension always continuous? Such questions have been studied, e.g., by D. Epstein and V. Markovic and more recently by S. Hurtado, Mann and E. Militon and automatic continuity proves to be a fundamental tool in this context.

(4) Geometric group theory: Geometric group theory or large scale geometry of finitely generated groups stimulated primarily by the work of Gromov is currently an area of high activity and interest, which permeates many areas of mathematics. Over the last 15 years, the large scale geometry of locally compact groups has also come into focus, both in the setting of Lie groups and for totally disconnected groups (a book surveying this is in preparation by Y. de Cornulier and P. de la Harpe). Research in this area connects naturally with the study of harmonic analytic aspects of locally compact groups and $C^*$-algebra techniques, e.g., rigdity and antirigidity properties such as Kazhdan's property (T) and the Haagerup property as evidenced in work by B. Bekka, de la Harpe, Y. Shalom, R. Tessera and A. Valette among many others. Lately, under the impetus of papers by D. Fisher and G. Margulis and by U. Bader, A. Furman, T. Gelander and N. Monod, the study of linear and affine actions on Banach spaces more general than Hilbert space has received a substantial amount of attention leading to an expanded toolset of relevance to the broader class of Polish groups.

Whereas non-locally compact Polish groups do not immediately lend themselves to the techniques of geometric group theory, recent discoveries by Rosendal allow one to transfer a large part of the machinery from the discrete or locally compact setting to all Polish groups without exception. This provides a unified conceptual framework in which geometric group theory and geometric non-linear functional analysis, i.e., the non-linear analysis of Banach spaces, are different instances of a single category and reveals deeper coarse geometric features of general Polish groups.

(5) Other areas: Many other areas have a vested interest in the structure of Polish groups or provide ideas for its further development. Of specific examples, one can mention the general study of infinite closed permutation groups, i.e., closed subgroups of the group all permutations of an infinite set, and its connections with model theory represented by researchers such as P. Cameron, D. Macpherson, P. Neumann, K. Tent and S. Thomas. Similarly, the permutation groups among locally compact groups are exactly the totally disconnected locally compact groups, a topic that has received substantial attention over the last 20 years through seminal work of G. Willis and was recently the subject of an Arbeitsgemeinschaft in Oberwolfach organised by P.-E. Caprace and Monod. And, finally, infinite-dimensional Lie groups, which from the pure side has received considerable attention, e.g., by H. Glockner and K.-H. Neeb.

(6) Workshop program: As evidenced by the flourishing activity in a broad range of areas, Polish groups is a vibrant field. Though partially represented in various conferences organised by different communities, thus far there has been no conference whose principal focus is the structure of Polish groups. Given the maturity of the subject and the unifying threads identified above, we feel that the time is ripe for a conference dedicated to this particular topic.

With the aim of ensuring the interdisciplinarity of the conference and to facilitate collaboration and transfer of knowledge between the different research groups, the intended participants would be selected from many different areas of mathematics in which Polish groups play a central role. Of course, as Polish groups encompass, e.g., all compact Lie groups and separable Banach spaces, some restriction is in order. As criteria for inclusion, we would select researchers whose prime interests are not entirely covered by the structure theory of classical, i.e., finite-dimensional Lie groups or whose concerns are closely related to the problems mentioned above. Also, several other factors have entered in identifying a list of potential participants, namlely, geographical variation and a proper represenation of women scholars. Similarly, we have focused on including younger people, students and post-docs in the line-up.