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Simplicial nonpositive curvature. (English) Zbl 1143.53039
Summary: We introduce a family of conditions on a simplicial complex that we call local $$k$$-largeness ($$k\geq 6$$ is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that $$k$$-largeness implies non-positive curvature if $$k$$ is sufficiently large. We also show that locally $$k$$-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C22 Geodesics in global differential geometry 53C20 Global Riemannian geometry, including pinching 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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