论文标题
超延伸空间的自动形态群体的密集局部有限的亚组
Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces
论文作者
论文摘要
我们通过证明Hall的通用局部有限基团可以嵌入urysohn空间的等轴测组以及随机图的自动形态组中，来验证Vershik的猜想。实际上，我们对所有已知无限超延伸空间的自动形态群体都显示了同样的情况。此外，其中包括有理Urysohn空间的等轴测组，超级Urysohn空间的等轴测组以及全部$ n \ geq 3 $的通用$ k_n $ free Graph的自动形态组。此外，我们表明在有限的度量空间或有限的关系结构上构成了有限的小组行动，形成了Fraïssé阶级，霍尔的群体是Fraïssé限制的代理小组。我们还将许多非同态可计数局部有限的局部有限基团嵌入到各种Urysohn空间的等轴测组中，并表明这些组的所有密集可计数亚组都是无混合的身份（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 $Δ$.
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 $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or 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 countable universal 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 $Δ$.
