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$F$isafinitegroupand$H$ a one-ended finitely presented group. Our main result is 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 $\mathrm{\mid }{F}_1\mathrm{\mid },\mathrm{\mid }{F}_2\mathrm{\mid }$ have the same prime divisors, or (ii) $H_1,H_2$ are amenable, $\mathrm{\mid }{F}_1\mathrm{\mid }=k^{n_1}$ and $\mathrm{\mid }{F}_2\mathrm{\mid }=k^{n_2}$ for some $k,n_1,n_2\ge 1$, and there exists a quasi-$(n_2\mathrm{/}{n}_1)$-to-one quasi-isometry $H_1\to H_2$. The article also contains algebraic information on groups quasi-isometric to such wreath products. 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.