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Hyperbolic rank and subexponential corank of metric spaces | | Sergei Buyalo
; Viktor Schroeder
; | | Date: |
14 Feb 2001 | | Subject: | Differential Geometry | math.DG | | 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 $g:X o T$ such that for each $tin T$ 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 $hyprank Xlesubcorank X$ 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 $hyp^n o X$ of the standard hyperbolic space $hyp^n$ with $n-1>dim X-
ank X$. | | Source: | arXiv, math.DG/0102109 | | Services: | Forum | Review | PDF | Favorites | |
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