Lamplighter-like geometry of groups

TL;DR

This paper introduces halo products, a generalization of wreath products, and develops a geometric framework to analyze their large-scale geometry, providing invariants to distinguish different halo groups up to quasi-isometry.

Contribution

The paper defines halo products and establishes a geometric framework for their analysis, extending the understanding of wreath-like group constructions.

Findings

Halo products generalize wreath products and include various group types.

A geometric framework with refined invariants is developed for halo groups.

The framework distinguishes halo groups up to quasi-isometry.

Abstract

In this article, we introduce halo products as a natural generalisation of wreath products. They also encompass lampshuffler groups $\mathrm{FSym}(H) \rtimes H$ and lampcloner groups $\mathrm{FGL}(H) \rtimes H$, as well as many possible variations based for instance on braid groups, Thompson's groups, mapping class groups, automorphisms of free groups. We build a geometric framework that allows us to study the large-scale geometry of halo groups, providing refined invariants distinguishing various halo groups up to quasi-isometry.