Asymptotic geometry of lamplighters over one-ended groups

This article is dedicated to the asymptotic geometry of wreath products F≀H:=(⨁_HF)⋊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.