Groups acting on horocyclic products

TL;DR

This paper classifies groups acting on horocyclic products of CAT(-κ) spaces, showing they are either ascending HNN extensions of virtually nilpotent groups or not finitely presented, based on boundary connectivity.

Contribution

It provides a classification of groups acting geometrically on horocyclic products of CAT(-κ) spaces, linking boundary properties to algebraic structure.

Findings

Groups are either ascending HNN extensions of virtually nilpotent groups or not finitely presented.

The classification depends on the connectivity of the visual boundary of the horocyclic product.

The work connects geometric boundary properties with algebraic group structures.

Abstract

Horocyclic products are a well-studied class of metric spaces that provide models for various solvable Lie groups, Baumslag-Solitar groups, and Lamplighter groups. Let $G$ act geometrically on a horocyclic product $X \bowtie Y$ of $\CAT(-\kappa)$ spaces $X,Y$. We show that every such group is either an ascending HNN extension of a finitely-generated virtually nilpotent group, or else is not finitely presented, depending on the connectivity of the visual boundary of $X\bowtie Y$.