What is a horocyclic product, and how is it related to lamplighters?

Abstract

This is a rather personal introductory outline of an interesting class of geometric, resp. graph- and group-theoretical structures. After an introductive section about their genesis, the general construction of horocyclic products is presented. Three closely related basic structures of this type are explained in more detail: Diestel-Leader graphs, treebolic spaces, and Sol-groups, resp. -manifolds. Emphasis is on their geometry, isometry groups, quasi-isometry classification and boundary at infinity. Subsequently, it is clarified under which parametrisation they admit discrete groups of isometries acting with compact quotient. Finally, further develpoments are reviewed briefly. Remark: the posted version contains a correction of a misprint that appeared on the bottom of page 22 in the version which appeared in IMN.