An illustrated introduction to the coarse topology of lamplighters

Roughly speaking, lamplighter graphs encode the possible configurations of a lamplighter that moves along a given graph and that modifies the colours of lamps at vertices. This article is dedicated to the following delicate question: when do two lamplighter graphs have the same coarse geometry, i.e. when are they quasi-isometric? Inspired by elementary ideas from topology, which we will "coarsify", I will survey some techniques that allow us to compare efficiently lamplighter graphs (and more) up to quasi-isometry. Based on a minicourse given during the Journées de Topologie Géométrique at the Institut Fourier in August 2025.