Simplicial nonpositive curvature

Januszkiewicz, Tadeusz; Świątkowski, Jacek

Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 1-85.

We introduce a family of conditions on a simplicial complex that we call local $k$-largeness ($k\ge 6$ 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.

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