Geometric & Functional Analysis GAFA

, Volume 12, Issue 2, pp 293–306

Hyperbolic rank and subexponential corank of metric spaces

  • S. Buyalo
  • V. Schroeder

DOI: 10.1007/s00039-002-8247-7

Cite this article as:
Buyalo, S. & Schroeder, V. GAFA, Geom. funct. anal. (2002) 12: 293. doi:10.1007/s00039-002-8247-7

Abstract.

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map \( g : X \to T \), T is a topological space, such that for each \( t \in T \) the set g-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality \( \textrm{rank}_h\,X \le\,\textrm{corank}\,X \) for a large class of metric spaces X including all locally compact Hadamard spaces, where rankhX is the maximal topological dimension of \( \partial_\infty Y \) among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rankh conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding \( H^n \to X \) of the standard hyperbolic space Hn with \( n - 1 > \textrm{dim}\,X - \textrm{rank}\,X \).

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • S. Buyalo
    • 1
  • V. Schroeder
    • 2
  1. 1.Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia, e-mail: buyalo@pdmi.ras.ruRU
  2. 2.Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland, e-mail: vschroed@math.unizh.chCH