Related papers: Polish groupoids and functorial complexity

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups,…

The following will be shown: Let $I$ be a $\sigma$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence…

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this…

We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups. The applications of our main…

The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for…

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 provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $\lambda$ of countable confinality. Somewhat unexpectedly, the resulting theory is…

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier…

We study the question of which Polish groups can be realized as subgroups of the unitary group of a separable infinite-dimensional Hilbert space. We also show that for a separable unital C$^*$-algebra $A$, the identity component…

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of…

We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem…

In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also…

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…

Let $G$ be a Polish group and let $H \leq G$ be a compact subgroup. We prove that there exists a Borel set $T \subset G$ which is simultaneously a complete set of coset representatives of left and right cosets, provided that a certain index…

For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We…

We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.

We consider various notions of equivalence in the space of bounded operators on a Hilbert space, in particular modulo finite rank, modulo Schatten $p$-class, and modulo compact. Using Hjorth's theory of turbulence, the latter two are shown…

In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for…

We study the equivalence classes under $\Delta^1_1$ isomorphism, otherwise effective-Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and we show the existence of strictly increasing and…

Given a general finite group $G$, there are various finite categories whose cohomology theories are of great interests. Recently Balmer and Grodal gave some new characterizations of the groups of endotrivial modules, via \v{C}ech cohomology…