Dense locally finite subgroups of automorphism groups of ultraextensive spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as dense subgroup in the isometry Urysohn space and automorphism random graph. In fact, we show same for all groups known infinite ultraextensive spaces. These include, addition, rational space, ultrametric spaces, Kn-free graph n≥3. Furthermore, actions on metric spaces or some relational structures form Fraïssé class, where appears acting limit. also embed continuum many non-isomorphic into various subgroups these are mixed identity free (MIF). Finally, give characterization isomorphism type Δ-metric terms distance value set Δ.