Regular maps from the lamplighter to metabelian groups (Q6647814)

The authors explore the concept of a \textit{regular map} (see [\textit{I. Benjamini} et al., Groups Geom. Dyn. 6, No. 4, 639--658 (2012; Zbl 1255.05074)]). In particular, let \(A\) and \(B\) be simplicial graphs with bounded degree. A map \(\phi: A \to B\) is said to be \textit{regular} if it satisfies the following conditions:\N\begin{itemize}\N\item[1.] There exists a constant \(K > 0\) such that \(d_B(\phi(x), \phi(y)) \leq K , d_A(x, y)\) for all \(x, y \in A\) (Lipschitz condition).\N\item[2.] There exists a constant \(C \geq 1\) such that \(|\phi^{-1}(b)| \leq C\) for every \(b \in B\).\N\end{itemize}\N\NThe authors work on the question posed by \textit{D. Hume} et al. [Geom. Funct. Anal. 32, No. 5, 1063--1133 (2022; Zbl 07605391)] concerning the existence of regular maps from the lamplighter group \(\mathbb{Z}_2 \wr \mathbb{Z}\) to finitely generated solvable groups with exponential growth. They present a positive result for metabelian groups, proving that there exists an injective regular map from \(\mathbb{Z}_2 \wr \mathbb{Z}\) to any finitely generated metabelian group with exponential growth.