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[Be]Benardete D., Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups.Trans. Amer. Math. Soc., 306 (1988), 499–527.
[BG]Bridson, M. &Gersten, S., The optimal isoperimetric inequality for torus bundles over the circle.Quart. J. Math. Oxford Ser. (2), 47 (1996), 1–23.
[BS1]Bieri, R. &Strebel, R., Almost finitely presented soluble groups.Comment. Math. Helv., 53 (1978) 258–278.
[BS2]—, Valuations and finitely presented metabelian groups.Proc. London Math. Soc., 41 (1980), 439–464.
[BW]Block, J. &Weinberger, S., Large scale homology theories and geometry, inGeometric Topology (Athens, GA, 1993), pp. 522–569. AMS/IP Stud. Adv. Math. 2.1. Amer. Math. Soc., Providence, RI, 1997.
[D]Dioubina, A., Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups.ar Xiv: math. GR/9911099.
[E]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. &Thurston, W. P.,Word Processing in Groups. Jones and Bartlett, Boston, MA, 1992.
[FJ1]Farrell, F. T. &Jones, L. E., A topological analogue of Mostow's rigidity theorem.J. Amer. Math. Soc., 2 (1989), 257–370.
[FJ2]—, Compact infrasolvmanifolds are smoothly rigid, inGeometry from the Pacific Rim (Singapore, 1994), pp. 85–97. de Gruyter, Berlin, 1997.
[FM1]Farb, B. &Mosher L., A rigidity theorem for the solvable Baumslag-Solitar groups.Invent. Math., 131 (1998), 419–451.
[FM2]— Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II.Invent Math., 137 (1999), 613–649.
[FM3]Farb, B. & Mosher, L., The geometry of surface-by-free groups. In preparation.
[FM4]—, Problems on the geometry of finitely generated solvable groups, inCrystallographic Groups and Their Generalizations (Kortrijk, 1999). Contemp. Math., 262. Amer. Math. Soc., Providence, RI, 2000.
[FS]Farb, B. &Schwartz, R., The large-scale geometry of Hilbert modular groups.J. Differential Geom., 44 (1996), 435–478.
[Ge1]Gersten, S. M., Quasi-isometry invariance of cohomological dimension.C. R. Acad. Sci. Paris Sér. I. Math., 316 (1993), 411–416.
[Ge2]— Isoperimetric functions of groups and exotic cohomology, inCombinatorial and Geometric Group Theory (Edinburgh, 1993), pp. 87–104. London Math. Soc. Lecture Note Ser., 204. Cambridge Univ. Press, Cambridge, 1995.
[GH]Ghys, E. &Harpe, P. De La, Infinite groups as geometric objects (after Gromov), inErgodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989), pp. 299–314. Oxford Univ. Press, New York, 1991.
[Gr1]Gromov, M., Groups of polynomial growth and expanding maps.Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73.
[Gr2]— Asymptotic invariants of infinite groups, inGeometric Group Theory, Vol. 2 (Sussex, 1991), pp. 1–295. London Math. Soc. Lecture Note Ser., 182 Cambridge Univ. Press, Cambridge, 1993.
[He]Heintze, E., On homogeneous manifolds of negative curvature.Math. Ann., 211 (1974), 23–24.
[Hi]Hinkkanen, A., Uniformly quasisymmetric groups.Proc. London Math. Soc., 51 (1985), 318–338.
[HPS]Hirsch, M., Pugh, C. &Shub, M.,Invariant Manifolds. Lecture Notes in Math., 583. Springer-Verlag, Berlin-New York, 1977.
[KK]Kapovich, M. & Kleiner, B., Coarse Alexander duality and duality groups. Preprint.
[Ma]Malcev, A. I., On a class of homogeneous spaces.Izv. Akad. Nauk SSSR Ser. Mat., 13 (1949), 9–32 (Russian); English translation inAmer. Math. Soc. Transl., 39 (1951), 1–33.
[Mi]Milnor, J., A note on curvature and fundamental group.J. Differential Geom., 2 (1968), 1–7.
[Mo1]Mostow, G. D., Factor spaces of solvable groups.Ann. of Math., 60 (1954), 1–27.
[Mo2]—Strong Rigidity of Locally Symmetric Spaces. Ann. of Math. Stud., 78. Princeton Univ. Press, Princeton, NJ, 1973.
[MSW]Mosher, L., Sageev, M. & Whyte, K., Quasi-actions on trees. In preparation.
[P1]Pansu, P., Dimension conforme et sphère à l'infini des variété à courbure négative.Ann. Acad. Sci. Fenn. Ser. A I Math., 14 (1989), 177–212.
[P2]—, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. of Math., 129 (1989), 1–60.
[S]Schwartz, R. E., Quasi-isometric rigidity and Diophantine approximation.Acta Math., 177 (1996), 75–112.
[T]Tukia, P., On quasi-conformal groups.J. Analyse Math., 46 (1986), 318–346.
[W]Witte, D., Topological equivalence of foliations of homogeneous spaces.Trans. Amer. Math. Soc., 317 (1990), 143–166.
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The first author was supported in part by NSF Grant DMS 9704640, by IHES and by the Alfred P. Sloan Foundation. The second author was supported in part by NSF Grant DMS 9504946 and by IHES.
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Farb, B., Mosher, L. On the asymptotic geometry of abelian-by-cyclic groups. Acta Math. 184, 145–202 (2000). https://doi.org/10.1007/BF02392628
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DOI: https://doi.org/10.1007/BF02392628