The Large Scale Geometry of the Higher Baumslag-solitar Groups

BS(m,n) =< x, y|xyx = y > 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. The first step towards a geometric understanding of the Baumslag-Solitar groups is to decide which among the BS(m,n) are quasi-isometric. The groups BS(1, n) are solvable, hence amenable, and so are not quasi-isometric to any of the BS(m,n) with 1 < m ≤ n which contain free subgroups and hence are are nonamenable. The solvable groups BS(1, n) are in many respects the most lattice-like of the Baumslag-Solitar groups. They are discrete subgroups in products of real and p-adic Lie groups. The groups BS(1, n) are classified up to quasi-isometry by Farb and Mosher in [FM1]. They prove that BS(1, n) and BS(1,m) are quasi-isometric only if n and m have common powers. When n and m have common powers BS(1, n) and BS(1,m) are not only quasi-isometric, but are commensurable (have isomorphic subgroups of finite index). This is the same rigidity phenomenon as occurs for nonuniform lattices in higher rank. Despite this rigidity, their full group of self quasiisometries is quite large, and in this they more closely resemble uniform lattices. In this paper we classify all the Baumslag-Solitar groups up to quasiisometry. The higher Baumslag-Solitar groups, namely those with 1 < m < n, are unlike the groups BS(1, n) in many ways. They are nonlinear, not residually finite, and usually not Hopfian. Indeed, this “bad” behavior was the motivation for their discovery. Our results show that the higher Baumslag-Solitar groups exhibit a surprising lack of rigidity; all the higher Baumslag-Solitar groups, aside from the degenerate case of BS(n, n), are quasi-isometric to each other. The quasi-isometries we construct do not reflect any clear algebraic relationship between the groups. In particular, many of the groups we prove to be quasi-isometric are not commensurable.