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

TL;DR

This paper introduces horocyclic products, exploring their geometric and group-theoretic properties, including classifications, isometry groups, and connections to lamplighter groups, with detailed analysis of structures like Diestel-Leader graphs and Sol-groups.

Contribution

It provides a comprehensive overview of horocyclic products, detailing their construction, geometric properties, and classifications, and clarifies their relation to lamplighter groups and associated structures.

Findings

Classification of horocyclic products based on geometry and isometry groups

Identification of conditions for discrete isometry groups with compact quotients

Analysis of boundary at infinity and quasi-isometry types

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.