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