Abstract
In this paper we provide the final steps in the proof of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin–Fisher–Whyte and Peng on the structure of quasiisometries of certain solvable Lie groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Auslander, An exposition of the structure of solvmanifolds, Bull. Amer. Math. Soc. (1973), 227–285.
Bridson M.R., Haefliger A.: Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin (1999)
Chow R.: Groups coarse quasi-isometric to complex hyperbolic space. Trans. Amer. Math. Soc. 348, 1757–1769 (1996)
Y. Cornulier, Dimension of asymptotic cones of Lie groups, J. of Topology, to appear.
A. Eskin, D. Fisher, K. Whyte, Quasi-isometries and rigidity of solvable groups, Pur. Appl. Math. Q., to appear.
A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries I: Rigidity for Sol and Lamplighter groups, to appear.
A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups, to appear.
Farb B., Mosher L.: Quasi-isometric rigidity for the solvable Baumslag–Solitar groups II. Invent. Math. 137, 613–649 (1999)
Farb B., Mosher L.: On the asymptotic geometry of abelian-by-cyclic groups. Acta Math. 184, 145–202 (2000)
B. Farb, L. Mosher, Problems on the geometry of finitely generated solvable groups, Crystallographic Groups and Their Generalizations (Kortrijk, 1999), Contemp. Math. 262, Amer. Math. Soc., Providence, RI (2000), 121–134.
Furman A.: Mostow–Margulis rigidity with locally compact targets. Geom. Funct. Anal. 11, 30–59 (2001)
Gersten S.M.: Quasi-isometry invariance of cohomological dimension. Comptes Rendues Acad. Sci. Paris Serie 1 Math. 316, 411–416 (1993)
F.P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand, 1969.
Guivarc’h Y.: Sur la loi des grands nombres et le rayon spectral dune marche aléatoire. Ast., Soc. Math. France 74, 47–98 (1980)
Mostow G.: Representative functions on discrete groups and solvable arithmetic subgroups. Amer. J. Math. 92, 1–32 (1970)
Osin D.: Exponential radicals of solvable Lie groups. J. Algebra 248, 790–805 (2002)
I. Peng, The quasi-isometry group of a subclass of solvable Lie groups I, preprint.
I. Peng, The quasi-isometry group of a subclass of solvable Lie groups II, preprint.
D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 97 (1981), 465–496.
Tukia P.: On quasiconformal groups. Journal d’Analyse Mathematique 46, 318–346 (1986)
Tyson J.: Metric and geometric quasiconformality in Ahlfors regular Loewner spaces Conform. Geom. Dyn. 5, 21–73 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by NSERC PGS B.
Rights and permissions
About this article
Cite this article
Dymarz, T. Large Scale Geometry of Certain Solvable Groups. Geom. Funct. Anal. 19, 1650–1687 (2010). https://doi.org/10.1007/s00039-010-0046-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-010-0046-y