Asymptotic geometry of lamplighters over one-ended groups

This article is dedicated to the asymptotic geometry of wreath products \(F\wr H := \left ( \bigoplus _{H} F \right ) \rtimes H\) 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 \(F\wr H\) 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 \(F_{1}\), \(F_{2}\) and two finitely presented one-ended groups \(H_{1}\), \(H_{2}\), we show that \(F_{1} \wr H_{1}\) and \(F_{2} \wr H_{2}\) are quasi-isometric if and only if either (i) \(H_{1}\), \(H_{2}\) are non-amenable quasi-isometric groups and \(|F_{1}|\), \(|F_{2}|\) have the same prime divisors, or (ii) \(H_{1}\), \(H_{2}\) are amenable, \(|F_{1}|=k^{n_{1}}\) and \(|F_{2}|=k^{n_{2}}\) for some \(k,n_{1},n_{2} \geq 1\), and there exists a quasi-\((n_{2}/n_{1})\)-to-one quasi-isometry \(H_{1} \to H_{2}\). This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of \(H=\mathbb{Z}\). 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.