Related papers: Polish G-spaces and continuous logic

We propose and study a new approach to the topologization of spaces of (possibly not all) future-directed causal curves in a stably causal spacetime. It relies on parametrizing the curves "in accordance" with a chosen time function. Thus…

In this paper we investigate the action of Polish groups (not necessary abelian) on an uncountable Polish spaces. We consider two main situations. First, when the orbits given by group action are small and the second when the family of…

We study continuous actions of Polish groups on Polish spaces. We develop Scott analysis introduced by Hjorth for studying orbit equivalence relations. We define eventually open actions and prove that this property characterizes the actions…

Suppose $G\curvearrowright X$ is a Polish group action, $H$ is a Polish group and $G\times X\overset{\psi}\longrightarrow H$ is a cocycle that is continuous in the second variable. If $\psi$ is either Baire measurable or is $\lambda\times…

We begin the systematic model theoretic study of $\mathrm{C}^*$-algebras using the tools of continuous logic.

We observe that a Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the…

The set of increasing functions on the rational numbers, equipped with the composition operation, naturally forms a topological semigroup with respect to the topology of pointwise convergence in which a sequence of increasing functions…

We extend the result of Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the general case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group…

In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical…

We prove that every open $\sigma$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in continuous logic. In particular, the orbit…

We apply the theory of large-scale geometry of Polish groups to groups of absolutely continuous homeomorphisms. Let $M$ be either the compact interval or circle. We prove that the Polish group $\operatorname{AC}_+(M)$ of…

We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $\sigma$-lacunary. In combination with [Invent. Math.201 (1), 309-383, 2015] we obtain a…

We show that a {\it Borel} action of a Polish group on a standard Borel space is Borel isomorphic to a {\it continuous} action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of…

Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces…

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly…

We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel…

We consider stability theory for Polish spaces and more generally for definable structures (say, with elements of a set of reals). We clarify by proving some equivalent conditions for $\aleph_0$-stability. We succeed to prove existence of…

We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $\cal K$ of spaces, and suppose that $\phi:K \to G$ is an abstract (i.e. not necessarily continuous)…

The first part of these notes give an introduction to the theory of Polish group actions on compact Hausdorff spaces, leading up to a proof of the Kechris-Pestov-Todorcevic correspondence and discussions of properties of universal minimal…

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.