Lamplighter-like geometry of groups

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.