Asymptotic geometry of lamplighters over one-ended groups (Q6614098)

Let \(F\) and \(H\) be two groups. The (restricted) wreath product \(F\wr H\) is defined as the semidirect product \(\big (\bigoplus_{H} F \big) \rtimes H\) where \(H\) acts on the direct sum by permuting the coordinates by left-multiplication.\N\NThe paper under review is dedicated to the asymptotic geometry of wreath products \(F \wr H\), where \(F\) is a finite group and \(H\) is a finitely generated group. The authors obtain a variety of interesting results.\N\NFirst, they prove that a coarse map from a finitely presented one-ended group to \(F \wr H\) must land at a bounded distance from a left coset of \(H\).\N\NLet \(f : X \rightarrow Y\) be a proper map between two graphs \(X\), \(Y\) and let \(\kappa > 0\). Then \(f\) is quasi-\(\kappa\)-to-one if there exists a constant \(C > 0\) such that \( \big | \kappa |A| - |f^{-1}(A)| \big |\) for all finite subset \(A \subset Y\).\N\NAs a second result, they 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 \(H_{1}\), \(H_{2}\), they 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 \(H_{1} \rightarrow H_{2}\). The latter result is an extension of two well-known papers by \textit{A. Eskin} et al. [Ann. Math. (2) 176, No. 1, 221--260 (2012; Zbl 1264.22005); Ann. Math. (2) 177, No. 3, 869--910 (2013; Zbl 1398.22012)], who treated the case of \(H=\mathbb{Z}\). Moreover, they, removing the one-ended assumption on the graphs, obtain Theorem 1.12: Let \(F_{1}\), \(F_{2}\) be two finite groups and \(H_{1}\), \(H_{2}\) two finitely presented groups. If \(F_{1}\wr H_{1}\) and \(F_{2} \wr H_{2}\) are quasi-isometric, then so are \(H_{1}\) and \(H_{2}\).\N\NAnother main result, which builds on the first, is a very restrictive description of quasi-isometries between two lamplighter groups on finitely presented one-ended groups. The paper contains many other results and in the final section, some open questions.