What is a horocyclic product, and how is it related to lamplighters? (Q2803889)

This is a survey of some work by the author and others on various structures associated with a question posed by the author in the 1980s about vertex-transitive graphs that do not look like Cayley graphs. It begins with some background on this question, and an answer provided by the `Diestel-Leader graphs' \(\mathrm{DL}(p,q)\), constructed by \textit{R. Diestel} and \textit{I. Leader} [J. Algebr. Comb. 14, No. 1, 17--25 (2001; Zbl 0985.05020)] and proved to be not quasi-isometric with any Cayley graph when \(p \neq q\), by \textit{A. Eskin} et al. [Ann. Math. (2) 176, No. 1, 221--260 (2012; Zbl 1264.22005)]. It then proceeds to describe the more general notions of Busemann functions and horocyclic products, and the special cases of `treebolic spaces' and Sol-groups and -manifolds, which share some geometric features with each other and with the Diestel-Leader graphs. Then it returns to the case \(p = q\) for the Diestel-Leader graphs, reporting on the fact (discovered by R.\ Möller and P.\ Neumann) that the graph \(\mathrm{DL}(p,p)\) is a Cayley graph, for a group that turned out to be the `lamplighter' group \(\mathbb{Z}_p \wr \mathbb{Z}\). The paper concludes with a review of some other developments.