Description:
<p>Let <italic>X</italic> be a proper 1-connected geodesic metric space which is non-positively curved in the sense that it satisfies Gromov’s <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CAT left-parenthesis 0 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mtext>CAT</mml:mtext>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\text {CAT}}(0)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> condition globally. If <italic>X</italic> is cocompact, then either it is <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta">
<mml:semantics>
<mml:mi>δ<!-- δ --></mml:mi>
<mml:annotation encoding="application/x-tex">\delta</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-hyperbolic, for some <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta greater-than 0">
<mml:semantics>
<mml:mrow>
<mml:mi>δ<!-- δ --></mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\delta > 0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, 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 <italic>K</italic> is not word-hyperbolic, then the universal cover of <italic>K</italic> contains a flat plane.</p>