Petersburg Department of Steklov Institute of Mathematics

PREPRINT 2/2001

HYPERBOLIC RANK AND SUBEXPONENTIAL CORANK OF METRIC SPACES

This preprint was accepted February 12, 2001
Contact: S.Buyalo

ABSTRACT:
We introduce a new quasi-isometry invariant $\subcorank X$
of a metric space $X$ called {\it subexponential corank}.
A metric space  $X$ has subexponential corank $k$
if roughly speaking there exists a continuous map $g:X\to T$
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
$\hyprank X\le\subcorank X$ for a large class of metric spaces $X$
including all locally compact Hadamard spaces, where $\hyprank X$
is maximal topological dimension of $\di Y$ 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 $\hyprank$ conjectured by M.Gromov, 
in particular, that any Riemannian symmetric space $X$
of noncompact type possesses no quasi-isometric embedding
$\hyp^n\to X$ of the standard hyperbolic space $\hyp^n$ with
$n-1>\dim X-\rank X$.