On the existence of flat planes in spaces of nonpositive curvature

Bridson, Martin R.

doi:10.1090/S0002-9939-1995-1273477-8

On the existence of flat planes in spaces of nonpositive curvature
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Proc. Amer. Math. Soc. 123 (1995), 223-235 Request permission

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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