Cross-wired lamplighter groups

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel–Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel–Leader graph admits a transitive, proper action on some other Diestel–Leader graph. We also give some examples of lattices that are not virtually lamplighters. This implies the class of discrete groups commensurable to lamplighter groups is not closed under quasiisometries and, combined with work of Eskin, Fisher and Whyte, gives a characterization of their quasi-isometry class.