Compact orbit spaces in Hilbert spaces and limits of edge-colouring models

Abstract

Let $G$ be a group of orthogonal transformations of a real Hilbert space $H$. Let $R$ and $W$ be bounded $G$-stable subsets of $H$. Let $\Vert .{\Vert }_R$ be the seminorm on $H$ defined by $\Vert x{\Vert }_R:={\sup }_{r\in R}|⟨r,x⟩|$ for $x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is compact for each $k\in\oN$, then the orbit space $W/G$ is compact when $W$ is equiped with the norm topology induced by $\Vert .{\Vert }_R$. As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lovász. It forms the edge-colouring counterpart of the graph limits of Lovász and Szegedy, which can be seen as limits of vertex-colouring models. In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called `spin models' and `vertex models' respectively.