Logic Seminar

Generalizing a Urysohn-like extension property for Hall's countable universal locally finite group, we define a concept of omnigenous groups and prove some results about such groups. One of the main results is that any countable omnigenous locally finite group can be embedded as a dense subgroup of the isometry group of the Urysohn space for all Δ-metric spaces, for any countable distance value set Δ. This implies a conjecture of Vershik from 2008. I will also talk about the current progress on the converse problem, namely to characterize all countable (locally finite) dense subgroups of the isometry groups of Urysohn spaces. This is joint work with Mahmood Etedadialiabadi, Francois Le Maître, and Julien Melleray.