Abstract We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case …
Abstract We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, …
We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of …
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of …
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, …
We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
Abstract We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case …
Abstract We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of …
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of …
We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.
Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made [Formula: see text] by refining the Polish topology on the …
Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made [Formula: see text] by refining the Polish topology on the real line. These sets are called potentially [Formula: see text]. We give a Hurewicz-like test to recognize potentially [Formula: see text] sets.
We study the extension of the Kechris-Solecki-Todorčević dichoto-my on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The …
We study the extension of the Kechris-Solecki-Todorčević dichoto-my on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.
Introduction A tree representation for Borel sets A double-tree representation for Borel sets Two applications of the tree representation Borel liftings of Borel sets More consequences and reverse results Proof …
Introduction A tree representation for Borel sets A double-tree representation for Borel sets Two applications of the tree representation Borel liftings of Borel sets More consequences and reverse results Proof of the main result Bibliography Index.
Let $\bfΓ$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leqω$ a cardinal. We study the Borel subsets of ${\mathbb R}^d$ that can be made …
Let $\bfΓ$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leqω$ a cardinal. We study the Borel subsets of ${\mathbb R}^d$ that can be made $\bfΓ$ by refining the Polish topology on the real line. These sets are called potentially $\bfΓ$. We give a test to recognize potentially $\bfΓ$ sets.
Ces travaux se situent dans le cadre de la theorie descriptive des ensembles. Je renvoie le lecteur a [Ku] (resp. [Mo]) pour les notions de base de theorie descriptive classique …
Ces travaux se situent dans le cadre de la theorie descriptive des ensembles. Je renvoie le lecteur a [Ku] (resp. [Mo]) pour les notions de base de theorie descriptive classique (resp. effective). Rappelons que dans le cas des espaces polonais de dimension 0, la hierarchie de Baire des boreliens est construite en alternant les operations de reunion denombrable et de passage au complementaire, en partant des ouverts-fermes, ce de maniere transfinie. On a alors la hierarchie suivante : Σ1 = ouverts Σ 0 2 = Fσ ... Σ 0 ω ... Π1 = fermes Π 0 2 = Gδ ... Π 0 ω ... On s’interesse ici a une hierarchie analogue a celle de Baire, sauf qu’au lieu de partir des ouvertsfermes d’un espace polonais de dimension 0, on part des produits de deux boreliens, chacun d’entre eux etant inclus dans un espace polonais. L’analogie devient plus claire quand on sait qu’etant donnes un espace polonais X et un borelien A de X , on peut trouver une topologie polonaise plus fine que la topologie initiale sur X (topologie ayant donc les memes boreliens), de dimension 0, et qui rende A ouvert-ferme. Pour notre probleme, le fait de travailler dans les espaces de dimension 0 n’est donc pas une restriction reelle. La definition qui suit apparait alors naturelle :
In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set <italic>A</italic> is both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" …
In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set <italic>A</italic> is both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript xi Overscript 0 Endscripts"> <mml:semantics> <mml:munderover> <mml:mo>∏<!-- ∏ --></mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:munderover> <mml:annotation encoding="application/x-tex">\prod _\xi ^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Delta _1^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it possesses a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript xi Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Pi _\xi ^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-code which is also <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Delta _1^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a by-product of the induction used to prove this result, we also obtain a separation result for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma _1^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sets: If two <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma _1^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sets can be separated by a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript xi Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Pi _\xi ^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> set, they can also be separated by a set which is both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Delta _1^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi Subscript xi Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Pi _\xi ^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.
xEy ¢} f(x) = f(y); so this means that there exist Borel calculable "invariants" f(x), belonging to some Polish space, thus of a fairly "concrete" nature, associated with each x …
xEy ¢} f(x) = f(y); so this means that there exist Borel calculable "invariants" f(x), belonging to some Polish space, thus of a fairly "concrete" nature, associated with each x E X which classify x up to E-equivalence.A typical example of such 2 classification is the case X = the n x n complex matrices (= en ), E = the equivalence relation of similarity between n x n matrices, and f(A) = the Jordan canonical form of A.
The $\mathbb {G}_0$-dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb {G}_0$-dichotomy for $\mathbf {\Sigma }^0_\xi$-measurable …
The $\mathbb {G}_0$-dichotomy due to Kechris, Solecki and Todorčević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb {G}_0$-dichotomy for $\mathbf {\Sigma }^0_\xi$-measurable countable colorings when $\xi \leq 3$. A $\mathbf {\Sigma }^0_\xi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\mathbf {\Sigma }^0_\xi$ squares. This leads to the study of countable unions of $\mathbf {\Sigma }^0_\xi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $\xi \leq 2$.
Let G be a permutation group on the set Ω. Usually we take |Ω| = ℵ0. We seek a notion of genericity for members of G. The idea is that …
Let G be a permutation group on the set Ω. Usually we take |Ω| = ℵ0. We seek a notion of genericity for members of G. The idea is that a member of G should be generic if it is in some sense 'typical'. We argue that the following is the correct definition. Suppose that G is endowed with a metric so that it becomes a complete metric space. It follows that the Baire category theorem holds, and we may use the notions of 'meagre' and 'comeagre' sets. An element of G is then said to be generic if it lies in a comeagre conjugacy class. The definition is given relative to the topology we have chosen. But in the main case, where G = Aut U for some countable relational structure U, G becomes a complete metric space on defining d(g, h) = Σ {2−n: xng ≠ xnh or xng −1 ≠ xnh−1} where {xn: n ε ο) is an enumeration of the domain of U, and we always take this metric in this case. In other instances we need to specify the topology explicitly. We study the circumstances under which generic elements do or do not exist. In particular we prove their existence in the following cases: G = Symm Ω, G = Aut Гc, with Гc the random C-coloured graph, and G = Aut (Q, >). Our notion of 'generic' is related to one studied by D. Lascar.
We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any …
We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any ω-stable ω-categorical structure, and (ii) the random graph have the small index property. We also show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally …
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally convex topological linear space. Day [3] has shown that amenability implies the fixed point property. For discrete groups he has shown the converse. Along with amenable groups, we shall study, in this paper, groups with the fixed point property. This paper is based on part of the author's Ph.D. dissertation at Yale University. The author wishes to express his thanks to his adviser, F. J. Hahn. NOTATION. Group will always mean topological group. For a group G, Go will denote the identity component. Likewise Ho will be the identity component of H, etc. Banach spaces, topological vector spaces, etc., will always be over the real field. For topology, we use the notation of Kelley [18], except that our spaces will always
The notion of a randomization of a first-order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new …
The notion of a randomization of a first-order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first-order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first-order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating …
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological …
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω-stable, ℵ0-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S∞ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We …
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We identify a minimal closed $G$-flow $I$ and an idempotent $r\in I$ (with respect to the Ellis se
Abstract We study automorphism groups of randomizations of separable structures, with focus on the ℵ 0 -categorical case. We give a description of the automorphism group of the Borel randomization …
Abstract We study automorphism groups of randomizations of separable structures, with focus on the ℵ 0 -categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ 0 -categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ 0 -categorical (except in basic cases).
Abstract We define a simple criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut( X ) satisfies all the main consequences of the existence …
Abstract We define a simple criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut( X ) satisfies all the main consequences of the existence of ample generics: it has the automatic continuity property, the small index property, and uncountable cofinality for nonopen subgroups. Then we verify it for the Urysohn space $&#xF094;$ , the Lebesgue probability measure algebra MALG, and the Hilbert space $\ell _2 $ , thus proving that Iso( $&#xF094;$ ), Aut(MALG), $U\left( {\ell _2 } \right)$ , and $O\left( {\ell _2 } \right)$ share these properties. We also formulate a condition for X which implies that every homomorphism of Aut( X ) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for $&#xF094;$ , and $\ell _2 $ .
On cherche à donner une construction aussi simple que possible d'un borélien donné d'un produit de deux espaces polonais. D'où l'introduction de la notion de classe de Wadge potentielle. On …
On cherche à donner une construction aussi simple que possible d'un borélien donné d'un produit de deux espaces polonais. D'où l'introduction de la notion de classe de Wadge potentielle. On étudie notamment ce que signifie "ne pas être potentiellement fer
We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the …
We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$ , due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.
In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with …
In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.
We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal. …
We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal's work as a guide, we explore the consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group $\operatorname {Iso}(\mathbf {U}_1)$ of the bounded Urysohn space, the unitary group ${\mathcal U}(\ell _2)$ of a separable Hilbert space, and the automorphism group $\operatorname {Aut}([0,1],\lambda )$ of the Lebesgue measure algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from $\operatorname {Aut}([0,1],\lambda )$ into a separable topological group is continuous.
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets Coanalytic sets …
Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets Coanalytic sets and pi 1/1 sets The Gandy-Harrington topology Polish Groups Metrics on topological groups Polish groups Continuity of homomorphisms The permutation group S Universal Polish groups The Graev metric groups Polish Group Actions Polish G-spaces The Vaught transforms Borel G-spaces Orbit equivalence relations Extensions of Polish group actions The logic actions Finer Polish Topologies Strong Choquet spaces Change of topology Finer topologies on Polish G-spaces Topological realization of Borel G-spaces Theory of Equivalence Relations Borel Reducibility Borel reductions Faithful Borel reductions Perfect set theorems for equivalence relations Smooth equivalence relations The Glimm-Effros Dichotomy The equivalence relation E0 Orbit equivalence relations embedding E0 The Harrington-Kechris-Louveau theorem Consequences of the Glimm-Effros dichotomy Actions of cli Polish groups Countable Borel Equivalence Relations Generalities of countable Borel equivalence relations Hyperfinite equivalence relations Universal countable Borel equivalence relations Amenable groups and amenable equivalence relations Actions of locally compact Polish groups Borel Equivalence Relations Hypersmooth equivalence relations Borel orbit equivalence relations A jump operator for Borel equivalence relations Examples of Fsigma equivalence relations Examples of pi 0/3 equivalence relations Analytic Equivalence Relations The Burgess trichotomy theorem Definable reductions among analytic equivalence relations Actions of standard Borel groups Wild Polish groups The topological Vaught conjecture Turbulent Actions of Polish Groups Homomorphisms and generic ergodicity Local orbits of Polish group actions Turbulent and generically turbulent actions The Hjorth turbulence theorem Examples of turbulence Orbit equivalence relations and E1 Countable Model Theory Polish Topologies of Infinitary Logic A review of first-order logic Model theory of infinitary logic Invariant Borel classes of countable models Polish topologies generated by countable fragments Atomic models and Gdelta-orbits The Scott Analysis Elements of the Scott analysis Borel approximations of isomorphism relations The Scott rank and computable ordinals A topological variation of the Scott analysis Sharp analysis of S -orbits Natural Classes of Countable Models Countable graphs Countable trees Countable linear orderings Countable groups Applications to Classification Problems Classification by Example: Polish Metric Spaces Standard Borel structures on hyperspaces Classification versus nonclassification Measurement of complexity Classification notions Summary of Benchmark Equivalence Relations Classification problems up to essential countability A roadmap of Borel equivalence relations Orbit equivalence relations General SIGMA 1/1 equivalence relations Beyond analyticity Appendix: Proofs about the Gandy-Harrington Topology The Gandy basis theorem The Gandy-Harrington topology on Xlow References Index
The subject of this book is the study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point …
The subject of this book is the study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces. These are equipped with canonical topological actions that give rise to the usual concepts of conjugacy of actions and cohomology of cocycles. Structural properties of discrete groups such as amenability, Kazhdan's property (T) and the Haagerup Approximation Property play a significant role in this theory as they have important connections to the global structure of these spaces. One of the main topics discussed in this book is the analysis of the complexity of the classification problems of conjugacy and orbit equivalence of actions, as well as of cohomology of cocycles. This involves ideas from topological dynamics, descriptive set theory, harmonic analysis, and the theory of unitary group representations. Also included in this title is a study of properties of the automorphism group of a standard probability space and some of its important subgroups, such as the full and automorphism groups of measure preserving equivalence relations and connections with the theory of costs. The book contains nine appendices that present necessary background material in functional analysis, measure theory, and group representations, thus making the book accessible to a wider audience.
Nous donnons, pour une certaine catégorie de boréliens d'un produit de deux espaces polonais, comprenant les boréliens à coupes dénombrables, une caractérisation du type "test d'Hurewicz" de ceux ne pouvant …
Nous donnons, pour une certaine catégorie de boréliens d'un produit de deux espaces polonais, comprenant les boréliens à coupes dénombrables, une caractérisation du type "test d'Hurewicz" de ceux ne pouvant pas être rendus différence transfinie d'ouverts
Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas …
Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas être rendus Γ par changement des deux topologies polo
Résumé: On donne des caractérisations des boréliens potentiellement d’une classe de Wadge donnée, parmi les boréliens à coupes verticales dénombrables d’un produit de deux espaces polonais. Pour ce faire, on …
Résumé: On donne des caractérisations des boréliens potentiellement d’une classe de Wadge donnée, parmi les boréliens à coupes verticales dénombrables d’un produit de deux espaces polonais. Pour ce faire, on utilise des résultats d’uniformisation partielle.
In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we …
In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].