A group is left-orderable if it admits a strict total order that is left-invariant under the group operation. The space of left-orderings of a given countable group is a well studied compact Polish space whose topological and dynamical features interact with the algebraic properties of the group. In this talk I will discuss the Borel complexity of the conjugacy equivalence relation on the spaces of left-orderings. This is joint work with Adam Clay.[-]

Generalized descriptive set theory has mostly been developed for uncountable cardinals satisfying the condition $\kappa ^{< \kappa }=\kappa$ (thus in particular for $\kappa$ regular). More recently the case of uncountable cardinals of countable cofinality has attracted some attention, partially because of its connections with very large cardinal axioms like I0. In this talk I will survey these recent developments and propose a unified approach which potentially could encompass all possible scenarios (including singular cardinals of arbitrary cofinality).[-]

I will talk about my result joint with S. Shelah establishing that the Borel space of torsion-free Abelian groups with domain ω is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. After this I will survey some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups, and on the problem of classification of countable co-Hopfian abelian and 2-nilpotent groups.[-]

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We give several applications of this result. We extend the Lyons-Nazarov theorem by showing that a bipartite Cayley graph admits a factor of iid perfect matching if and only if the group is not iso-morphic to the semidirect product of Z and a finite group of odd order, answering a question of Kechris and Marks in the bipartite case. We also answer an open question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we show how our results generalize and lead to a simple approach to recent results on measurable circle squaring.[-]