Vertex-transitive median graphs of non-exponential growth

We characterize vertex-transitive median graphs of non-exponential growth as the Cartesian products of finite hypercubes with finite dimensional lattice graphs. Additionally, we prove that every median graph without convex subgraphs isomorphic to K1,3K1,3 or the 4-pan graph is isomorphic to the weak Cartesian product of finite paths, rays and two way infinite paths.