Simplicial nonpositive curvature

Januszkiewicz, Tadeusz; Świątkowski, Jacek

doi:10.1007/s10240-006-0038-5

Geodesic

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Simplicial nonpositive curvature

Januszkiewicz, Tadeusz ; Świątkowski, Jacek

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

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Résumé

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.

DOI : 10.1007/s10240-006-0038-5

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     author = {Januszkiewicz, Tadeusz and \'Swi\k{a}tkowski, Jacek},
     title = {Simplicial nonpositive curvature},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--85},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     doi = {10.1007/s10240-006-0038-5},
     mrnumber = {2264834},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-006-0038-5/}
}

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Januszkiewicz, Tadeusz; Świątkowski, Jacek. Simplicial nonpositive curvature. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 1-85. doi: 10.1007/s10240-006-0038-5

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