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

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 $||\cdot||_R$ be the seminorm on $H$ defined by $||x||_R := sup_{r\in R}|\langle r,x\rangle|$, for $x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is compact for each $k\in \mathbb{N}$, then the orbit space $W/G$ is compact when $W is equiped with the norm topology induced by $||\cdot||_R$. As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lov\'asz. It forms the edge-colouring counterpart of the graph limits of Lov\'asz 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.