The bi-embeddability relation for finitely generated groups II

We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups

The bi-embeddability relation for finitely generated groups

There does not exist a Borel selection of an isomorphism class within each bi-embeddability class of finitely generated groups

The universality of polynomial time Turing equivalence

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees

Universal countable Borel quasi-orders

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders. In this paper we are concerned with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, and then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees. Our main result states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order. Finally, we show that, up to Borel bireducibility, there are continuum-many distinct countable Borel quasi-orders which symmetrize to a universal countable Borel equivalence relation

The bi-embeddability relation for finitely generated groups II

We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups.The final publication is available at Springer via http://dx.doi.org/10.1007/s00153-015-0455-6.Peer reviewe