# Hyperbolic rank and subexponential corank of metric spaces

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

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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 \( g : X \to T \), *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 dim*T* = *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 rank_{h}*X* 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 rank_{h} 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 H^{n} with \( n - 1 > \textrm{dim}\,X - \textrm{rank}\,X \).