Pattern preserving quasi-isometries in lamplighter groups and other related groups

Abstract

In this paper we explore the interplay between aspects of the geometry and algebra of three families of groups of the form B semidirect the integers 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 affine maps of B. We show that this is no longer true in the lamplighter case but the induced maps do share some features with affine maps.