Polish groups and Baire category methods
type: Item
instance of: scholarly article
title: Polish groups and Baire category methods
zbMATH Open document ID: 1364.22002
author: Julien Melleray
published in: Confluentes Mathematici
publication date: 2017-02-03T00:00:00Z
review text: The article is a~modified version of the author's habilitation thesis, presenting his work on topics related to Polish groups, Baire category methods and metric model theory. Some of the results are new. In particular, it is shown that for any countably infinite group~\(G\) all conjugacy classes in the space of actions of~\(G\) on the Urysohn space are meager and that the group of bounded isometries of the Urysohn space endowed with the topology of uniform convergence is path-connected. The work consists of five sections. The first section is an introduction. In the second section, some concepts of Baire category theory and Polish groups are recalled. The third section is devoted to the Urysohn space~\(\mathbf{U}\) and the isometry group of~\(\mathbf{U}\). There is only one way of embedding~\(\mathbf{U}\) isometrically into a~normed vector space. The spaces with this rigidity property are investigated. In the fourth section, the Baire category in the space of actions of a~countable group on some homogeneous structures is studied. This concerns mainly to the space of actions of a separable Hilbert space, of the standard atomless probability algebra, and of the Urysohn space. In the last fifth section, the language of first-order logic and metric model theory play a~crucial role in the investigation of Polish groups.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 22F50
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 6680206
zbMATH Keywords: Polish groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: Baire category
zbMATH Keywords: topological dynamics
zbMATH Keywords: minimal homeomorphisms
MaRDI profile type: Publication
full work available at URL: cml.28
OpenAlex ID: W2528139899
cites work: Quasi finitely axiomatizable totally categorical theories
cites work: Good measures on Cantor space
cites work: The Algebraic Simplicity of Certain Groups of Homeomorphisms
cites work: On embedding uniform and topological spaces
cites work: The infinite unitary and related groups are algebraically determined Polish groups
cites work: Q3227858
cites work: Q4331784
cites work: The linear isometry group of the Gurarij space is universal
cites work: Q3602612
cites work: Polish topometric groups
cites work: Generic orbits and type isolation in the Gurarij space
cites work: Grey subsets of Polish spaces
cites work: Continuous first order logic and local stability
cites work: Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems
cites work: Q5499954
cites work: Measure conjugacy invariants for actions of countable sofic groups
cites work: Some isometry groups of the Urysohn space
cites work: Commuting point transformations
cites work: Entropie d'un groupe ab�lien de transformations
cites work: The conjugacy problem for the automorphism group of the random graph
cites work: The generic transformation can be embedded in a flow.
cites work: Q3919918
cites work: Divisibility of countable metric spaces
cites work: Q3329946
cites work: Separable complete ANR's admitting a group structure are Hilbert manifolds
cites work: The Choquet simplex of invariant measures for minimal flows
cites work: On Groups of Measure Preserving Transformations. I
cites work: On Groups of Measure Preserving Transformations. II
cites work: Transformation groups and \(C^ *\)-algebras
cites work: Le groupe des transformations de \([0,1]\) qui preservent la mesure de Lebesgue est un groupe simple
cites work: Properties of the Holmes space
cites work: An anti-classification theorem for ergodic measure preserving transformations
cites work: Sur l'extension aux relations de quelques propriétés des ordres
cites work: An invariant for rigid rank-1 transformations
cites work: Q4315764
cites work: Q3525843
cites work: On the classification of Polish metric spaces up to isometry
cites work: TOPOLOGICAL GROUP STRUCTURES OF INFINITE SYMMETRIC GROUPS
cites work: SOME EXTREMELY AMENABLE GROUPS RELATED TO OPERATOR ALGEBRAS AND ERGODIC THEORY
cites work: Q4859247
cites work: Full groups of Cantor minimal systems
cites work: The automorphism group of the Gaussian measure cannot act pointwise
cites work: On minimal actions of Polish groups
cites work: The group \(\text{Aut}(\mu)\) is Roelcke precompact
cites work: WEAK ORBIT EQUIVALENCE OF CANTOR MINIMAL SYSTEMS
cites work: Topological groups with Rokhlin properties
cites work: Lipschitz-free Banach spaces
cites work: On algebraic properties of topological full groups
cites work: A Topological Application of the Isoperimetric Inequality
cites work: Q5552458
cites work: On Monothetic Groups
cites work: On the existence of pathological submeasures and the construction of exotic topological groups
cites work: Q4933738
cites work: An oscillation theorem for groups of isometries
cites work: Q4003410
cites work: The Small Index Property for ω‐Stable ω‐Categorical Structures and for the Random Graph
cites work: Q4020553

Polish groups and Baire category methods (cites work)
cites work: The Urysohn space embeds in Banach spaces in just one way

Polish groups and Baire category methods (cites work)
cites work: Urysohn universal space, its development and Hausdorff's approach

Polish groups and Baire category methods (cites work)
cites work: Full groups of minimal homeomorphisms and Baire category methods

Polish groups and Baire category methods (cites work)
cites work: Connected Polish groups with ample generics

Polish groups and Baire category methods (cites work)
cites work: A Uniqueness Result for Topological Groups

Polish groups and Baire category methods (cites work)
cites work: Uniqueness Results for Groups of Measure Perserving Transformations

Polish groups and Baire category methods (cites work)
cites work: Every reasonably sized matrix group is a subgroup of $S_∞$

Polish groups and Baire category methods (cites work)
cites work: Q5838558

Polish groups and Baire category methods (cites work)
cites work: Q3784822

Polish groups and Baire category methods (cites work)
cites work: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups

Polish groups and Baire category methods (cites work)
cites work: Q4323294

Polish groups and Baire category methods (cites work)
cites work: Q4411218

Polish groups and Baire category methods (cites work)
cites work: Q3407797

Polish groups and Baire category methods (cites work)
cites work: Turbulence, amalgamation, and generic automorphisms of homogeneous structures

Polish groups and Baire category methods (cites work)
cites work: Sofic measure entropy via finite partitions

Polish groups and Baire category methods (cites work)
cites work: Turbulence, representations, and trace-preserving actions

Polish groups and Baire category methods (cites work)
cites work: The commutant is the weak closure of the powers, for rank-1 transformations

Polish groups and Baire category methods (cites work)
cites work: The generic transformation has roots of all orders

Polish groups and Baire category methods (cites work)
cites work: Topological properties of full groups

Polish groups and Baire category methods (cites work)
cites work: A proof of uniqueness of the Gurariĭ space

Polish groups and Baire category methods (cites work)
cites work: The Gurarij spaces are unique

Polish groups and Baire category methods (cites work)
cites work: Ergodic theory and virtual groups

Polish groups and Baire category methods (cites work)
cites work: An example of a non non-Archimedean Polish group with ample generics

Polish groups and Baire category methods (cites work)
cites work: Reconstruction of orbits of Cantor systems from full groups

Polish groups and Baire category methods (cites work)
cites work: Linearly rigid metric spaces and the embedding problem

Polish groups and Baire category methods (cites work)
cites work: Some geometric and dynamical properties of the Urysohn space

Polish groups and Baire category methods (cites work)
cites work: A Note on Hjorth's oscillation theorem

Polish groups and Baire category methods (cites work)
cites work: Topology of the isometry group of the Urysohn space

Polish groups and Baire category methods (cites work)
cites work: Extensions of generic measure-preserving actions

Polish groups and Baire category methods (cites work)
cites work: Polish groups with metrizable universal minimal flows

Polish groups and Baire category methods (cites work)
cites work: Generic representations of abelian groups and extreme amenability

Polish groups and Baire category methods (cites work)
cites work: Topological semigroups and fixed points

Polish groups and Baire category methods (cites work)
cites work: Q4220920

Polish groups and Baire category methods (cites work)
cites work: The distortion problem

Polish groups and Baire category methods (cites work)
cites work: Q5677743

Polish groups and Baire category methods (cites work)
cites work: Invariant Measures in Groups Which are not Locally Compact

Polish groups and Baire category methods (cites work)
cites work: On free actions, minimal flows, and a problem by Ellis

Polish groups and Baire category methods (cites work)
cites work: Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups

Polish groups and Baire category methods (cites work)
cites work: Q3415741

Polish groups and Baire category methods (cites work)
cites work: Q3417469

Polish groups and Baire category methods (cites work)
cites work: On continuity and openess of homomorphisms in topological groups

Polish groups and Baire category methods (cites work)
cites work: Q3707983

Polish groups and Baire category methods (cites work)
cites work: On the non-existence of certain group topologies

Polish groups and Baire category methods (cites work)
cites work: Automatic Continuity of Group Homomorphisms

Polish groups and Baire category methods (cites work)
cites work: A topological version of the Bergman property

Polish groups and Baire category methods (cites work)
cites work: Automatic continuity of homomorphisms and fixed points on metric compacta

Polish groups and Baire category methods (cites work)
cites work: AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

Polish groups and Baire category methods (cites work)
cites work: Über die Permutationsgruppe der natürlichen Zahlenfolge

Polish groups and Baire category methods (cites work)
cites work: Sur un espace métrique séparable universel

Polish groups and Baire category methods (cites work)
cites work: Closed subgroups generated by generic measure automorphisms

Polish groups and Baire category methods (cites work)
cites work: Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation

Polish groups and Baire category methods (cites work)
cites work: A course in model theory

Polish groups and Baire category methods (cites work)
cites work: The isometry group of the bounded Urysohn space is simple

Polish groups and Baire category methods (cites work)
cites work: On the isometry group of the Urysohn space

Polish groups and Baire category methods (cites work)
cites work: Infinite products of finite simple groups II

Polish groups and Baire category methods (cites work)
cites work: Automatic continuity for the unitary group

Polish groups and Baire category methods (cites work)
cites work: Q3259049

Polish groups and Baire category methods (cites work)
cites work: Q3477652

Polish groups and Baire category methods (cites work)
cites work: The Urysohn universal metric space is homeomorphic to a Hilbert space

Polish groups and Baire category methods (cites work)
cites work: The infinite-dimensional topology of function spaces

Polish groups and Baire category methods (cites work)
cites work: Q4266131

Polish groups and Baire category methods (cites work)
cites work: Conjugacy class conditions in locally compact second countable groups

Polish groups and Baire category methods (cites work)
cites work: Canonical embeddings of compact metric spaces

Polish groups and Baire category methods (cites work)
cites work: Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem

Polish groups and Baire category methods (DOI)
DOI: 10.5802/CML.28