Asymptotic geometry of lamplighters over one-ended groups

• 成果类型: Article
• 署名作者: Genevois, Anthony; Tessera, Romain
• 署名单位: Universite de Montpellier; Sorbonne Universite; Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01278-w
• 发表日期: 2024
• 页码: 1-67
• 关键词: quasi-isometric equivalence; coarse differentiation; wreath-products; random-walks; bilipschitz equivalence; proper actions; RIGIDITY; compression; EMBEDDINGS; graphs

摘要

This article is dedicated to the asymptotic geometry of wreath products F H := H F H where F is a finite group and H is a finitely generated group. Our first main result says that a coarse map from a finitely presented one-ended group to F H must land at bounded distance from a left coset of H. Our second main result, building on the later, is a very restrictive description of quasi-isometries between two lamplighter groups on finitely presented one-ended groups. Third, we obtain a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups F1, F2 and two finitely presented one-ended groups H1, H2, we show that F1 H1 and F2 H2 are quasi-isometric if and only if either (i) H1, H2 are non-amenable quasi-isometric groups and |F1|, |F2| have the same prime divisors, or (ii) H1, H2 are amenable, |F1| = kn1 and |F2| = kn2 for some k, n1, n2 = 1, and there exists a quasi-(n2/ n1)-to-one quasi-isometry H1. H2. This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of H = Z. Our approach is however fundamentally different, as it crucially exploits the assumption that H is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces.