An illustrated introduction to the coarse topology of lamplighters

Anthony Genevois

Home/Archive/arXiv

arXiv · 2509.18941

An illustrated introduction to the coarse topology of lamplighters

Anthony Genevois

View original source ↗Read source document ↗Export metadata ↓

Abstract

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 Journ\'ees de Topologie G\'eom\'etrique at the Institut Fourier in August 2025.

Explore related subjects

math.GR math.GT math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Anthony Genevois. 2025-09-23. An illustrated introduction to the coarse topology of lamplighters. https://arxiv.org/abs/2509.18941

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗

arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗

arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗