We verify a conjecture of Vershik by showing that Hall’s universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Generalizing a Urysohn-like extension property for Hall's group, we introduce a notion of "omnigenous groups" and show that every locally finite omnigenous group can be embedded as a dense subgroup in the isometry groups of various Urysohn spaces.
This is joint work with Su Gao, Francois Le Maître, and Julien Melleray.