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 $n\ge 3$. Furthermore, we show that finite group actions on finite metric spaces or some finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic universal countable locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn Δ-metric spaces in terms of the distance value set Δ.

我们通过证明霍尔的普遍可数局部有限群可以作为稠密子群嵌入 Urysohn 空间的等距群和随机图的自同构群来验证 Vershik 的猜想。事实上，我们对已知无限超延展空间的所有自同构群都显示了相同的结果。此外，这些还包括有理 Urysohn 空间的等距群、超度量 Urysohn 空间的等距群和全称的自同构群。${钾}_n$- 所有人的免费图表 $n\ge 3$. 此外，我们证明了有限度量空间或一些有限关系结构上的有限群作用形成了 Fraïssé 类，其中 Hall 群​​表现为 Fraïssé 极限的作用群。我们还将连续统的许多非同构通用可数局部有限群嵌入到各种 Urysohn 空间的等距群中，并表明这些群的所有稠密可数子群都是混合恒等式（MIF）。最后，我们根据距离值集Δ给出了Urysohn Δ-度量空间的等距群的同构类型的表征。