On the complexity of embeddability between groups

Filippo Calderoni

Università di Torino

After a gentle introduction to the theory of Borel reducibility, we present some recent results about the complexity of quasi-orders between groups: we exploit the main techniques developed by Camerlo-Marcone-Motto Ros to show that the embeddability between countable groups and the topological embeddability between Polish groups are invariantly universal quasi-orders. In so doing, we strengthen a result obtained by Williams and a result by Ferenczi-Louveau-Rosendal. Further, if time permits, we discuss a work in progress on the complexity of the embeddability between groups of uncountable size.