Hyperbolic rank and subexponential corank of metric spaces |
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Authors: | S Buyalo V Schroeder |
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Institution: | Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia, e-mail: buyalo@pdmi.ras.ru, RU Institut für Mathematik, Universit?t Zürich, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland, e-mail: vschroed@math.unizh.ch, CH
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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 , T is a topological space, such that for each 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 for a large class of metric spaces X including all locally compact Hadamard spaces, where rank
h
X is the maximal topological dimension of 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 rank
h
conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H
n
with .
Submitted: February 2001, Revised: October 2001. |
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Keywords: | |
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