The large scale geometry of the higher Baumslag-Solitar groups

The Baumslag-Solitar groups: BS(m,n)=<x,y| x y^{m} x^{-1} = y^{n}> are some of the simplest interesting infinite groups which are not lattices in Lie groups. They have been studied in depth from the point of view of combinatorial group theory. It is natural to ask if the geometric approach to the theory of infinite groups, which has been so successful in the study of lattices, can yield any insights in this nonlinear case. We show that in contrast to the solvable groups studied by Farb and Mosher, the groups BS(m,n) with 1<m<n are all quasi-isometric. This generalizes to almost all groups acting on trees with infinite cyclic vertex and edge stabilizers.