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.
