Quasi-isometric Rigidity for the Solvable Baumslag-solitar Groups, Ii. Outline of the Paper Acknowledgements

Introduction Gromov's Polynomial Growth Theorem [Gro81] characterizes the class of virtually nilpotent groups by their asymptotic geometry. Since Gromov's theorem it has been a major open question (see, e.g. [GH91]) to find an appropriate generalization for solvable groups. This paper gives the first step in that direction. One fundamental class of examples of finitely-generated solvable groups which are not virtually nilpotent are the solvable Baumslag-Solitar groups BS(1, n) = a, b bab −1 = a n where n ≥ 2. Our main theorem characterizes the group BS(1, n) among all finitely-generated groups by its asymptotic geometry. Theorem A (Quasi-isometric rigidity). Let G be any finitely generated group. If G is quasi-isometric to BS(1, n) for some n ≥ 2, then there is a short exact sequence where N is finite and Γ is abstractly commensurable to BS(1, n). In fact we will describe the precise class of quotient groups Γ which can arise, and will classify all torsion-free G; see section 5 in the outline below. Theorem A complements the main theorem of [FM97], where it is shown that BS(1, n) is quasi-isometric to BS(1, m) if and only if they are abstractly 1 commensurable, which happens if and only if m, n are positive integer powers of the same positive integer. Theorem A says that every finitely generated group quasi-isometric to BS(1, n) can be obtained from BS(1, n) by first passing to some abstractly commensurable group and then to some finite extension. We describe this phenomenon by saying that the group BS(1, n) is quasi-isometrically rigid. This property is even stronger than what we know for nilpotent groups, for while Gromov's theorem says that the class of nilpotent groups is a quasi-isometrically rigid class, outside of a few low-dimensional cases it is not known whether an individual nilpotent group must always be quasi-isometrically rigid. Comparison with lattices Recent work on lattices in semisimple Lie groups has established the quasi-isometric classification of all such lattices. In the case of a nonuniform lattice Λ in a semisimple Lie group G = SL(2, R), quasi-isometric rigidity of Λ follows from the deep fact that the quasi-isometry group QI(Λ) is the commensurator group of Λ in G, a count-able group (see [Sch96b], [Sch96a], [FS96], [Esk96], or [Far96] for a survey). In contrast, for uniform lattices Λ in the isometry group of X = H n or CH n , the quasi-isometry group …