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 n – 1 > dim X – rank X.

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Buyalo, S; Schroeder, V (2002). *Hyperbolic rank and subexponential corank of metric spaces.* Geometric and Functional Analysis, 12(2):293-306.

## 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 n – 1 > dim X – rank X.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2002 |

Deposited On: | 29 Nov 2010 16:27 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Birkhäuser |

ISSN: | 1016-443X |

Additional Information: | The original publication is available at www.springerlink.com |

Publisher DOI: | 10.1007/s00039-002-8247-7 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1911661 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0992.54027 |

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