论文标题
规定的图形对称性和刚性的风味
Prescribed graphon symmetries and flavors of rigidity
论文作者
论文摘要
我们证明，一个任意的紧凑型Metrizable群体可以实现为图形的自动形态组。这是Frucht定理恢复任意有限群的连续类似物是有限图的自动形态组。 本文还包含许多结果，即在传递到图形限制时，紧凑型组动作的传播持续性。如果每当它对图形的收敛序列的每个成员$γ_n$进行传输时，请调用紧凑型组$ \ mathbb {g} $ graphon-rigid。我们表明，对于紧凑的谎言组$ \ mathbb {g} $ graphon刚度等效于标识组件$ \ mathbb {g} _0 $ as samisimple;作为与Lovász和Szegedy的结果的部分交谈，这也等同于弱随机性：该组在每个维度中仅具有有限的许多不可还原表示的属性。同样，如果每个紧凑型谎言组$ \ mathbb {h} $，请调用紧凑型组$ \ mathbb {g} $ image-rigid。我们证明，图形刚度意味着与连接或涂鸦的紧凑型组的图像刚度，并且这两个条件是等效的（也等同于扭转）。
We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphing; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups are automorphism groups of finite graphs. The paper also contains a number of results the persistence of transitivity of a compact-group action upon passing to a limit of graphons. Call a compact group $\mathbb{G}$ graphon-rigid if, whenever it acts transitively on each member $Γ_n$ of a convergent sequence of graphons, it also acts transitively on the limit $\lim_n Γ$. We show that for a compact Lie group $\mathbb{G}$ graphon rigidity is equivalent to the identity component $\mathbb{G}_0$ being semisimple; as a partial converse to a result of Lovász and Szegedy, this is also equivalent to weak randomness: the property that the group have only finitely many irreducible representations in each dimension. Similarly, call a compact group $\mathbb{G}$ image-rigid if for every compact Lie group $\mathbb{H}$ the images of morphisms $\mathbb{G}\to \mathbb{H}$ form a closed set (of closed subgroups, in the natural topology). We prove that graphon rigidity implies image rigidity for compact groups that are either connected or profinite, and the two conditions are equivalent (and also equivalent to being torsion) for profinite abelian groups.
