In this talk, I will explore the interplay between aspects of the geometry and algebra of three families of groups of the form $B\rtimes Z$, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in $SOL$. In particular we examine what kind of maps are induced on $B$ by quasi-isometries that coarsely permute cosets of the $Z$ subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in $SOL$ and solvable Baumslag-Solitar cases respectively such quasi-isometries induce parallelogram preserving maps of $B$. We show that this is no longer true in the lamplighter case but the induced maps do share some features with parallelogram preserving maps. This is joint work with Dymarz, Macura and Morris-Wright.