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 ofH. Let ‖.‖R be the seminorm onH defined by ‖x‖R := supr∈R |〈r, x〉| for x ∈ H. We show that if W is weakly compact and the orbit space R/G is compact for each k ∈ N, then the orbit space W/G is compact when W is equiped with the norm topology induced by ‖.‖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, vertexand edge-colouring models are called ‘spin models’ and ‘vertex models’ respectively. By relaxing the condition that W is bounded, also the compactness of the space of L graphons introduced by Borgs, Chayes, Cohn, and Zhao follows.