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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the existence of flat planes in spaces of nonpositive curvature


Author: Martin R. Bridson
Journal: Proc. Amer. Math. Soc. 123 (1995), 223-235
MSC: Primary 53C23; Secondary 20F32
MathSciNet review: 1273477
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Abstract: Let X be a proper 1-connected geodesic metric space which is non-positively curved in the sense that it satisfies Gromov's $ {\text{CAT}}(0)$ condition globally. If X is cocompact, then either it is $ \delta $-hyperbolic, for some $ \delta > 0$, or else it contains an isometrically embedded copy of the Euclidean plane; these conditions are mutually exclusive. It follows that if the fundamental group of a compact non-positively curved polyhedron K is not word-hyperbolic, then the universal cover of K contains a flat plane.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1273477-8
PII: S 0002-9939(1995)1273477-8
Article copyright: © Copyright 1995 American Mathematical Society