New York Journal of Mathematics
Volume 18 (2012) 667-677


Yves de Cornulier, David Fisher, and Neeraj Kashyap

Cross-wired lamplighter groups

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Published: September 30, 2012
Keywords: Tree, Busemann function, Diestel-Leader graph, double cosets
Subject: 05C63 (primary); 20E22 (secondary)

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 quasi-isometries and, combined with work of Eskin, Fisher and Whyte, gives a characterization of their quasi-isometry class.


D. F. was partially supported by NSF grant DMS 0643546

Author information

Yves de Cornulier:
Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France

David Fisher and Neeraj Kashyap:
Department of Mathematics, Indiana University - Bloomington, Rawles Hall, Bloomington, IN 47401, USA