On isometric actions

To a cardinal k ≥ 2, we associate a simply-connected polyhedral surface Σk endowed with a bounded metric dk such that every group of cardinality k has an isometric, properly discontinuous action on (Σk, dk). If ℵ0 ≤ k ≤ 2ℵ0 and G is a group of cardinality k, then we extend (Σk, dk) to a simply-connected bounded metric space (MG, dG) such that the action of G extends to an isometric, properly discontinuous action on (MG, dG) and G is the full isometry-group of (MG, dG).