12 pages. Since completing the first version, we have become aware that the main theorem is originally due to Gromov, and that the argument has been previously described by Alexandre Martin in the more general setting of small cancellation theory over graphs of groups (see http://arxiv.org/abs/1306.6847). We hope that our short account of the proof in the classical setting will still be useful
We give a proof that groups satisfying the "uniform C'(1/6)" small cancellation condition admit a geometric action on a CAT(-1) space. It follows that random groups at density <1/12 are CAT(-1). The proof consists of a direct construction of a piecewise hyperbolic structure on the presentation complex of such a group, together with folding moves to make the complex negatively curved. The argument was originally suggested by Gromov.
Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility
我们提出了一个度量条件${\LARGEτ}'$，它描述了经典的小取消组的几何形状，也适用于其他已知类的组，如二维Artin组。我们证明了满足条件${\LARGEτ}'$的演示文稿在Sieradski和Gersten的意义下是图解可约的。特别是我们推导出的标准表示的阿廷群是非球面的当且仅当它是图解可约。证明了在一些额外的假设下，${\LARGEτ}'$ -群具有二次Dehn函数和可解的共轭问题.格林林格引理的精神，我们证明，如果一个演示$P=\langle X \mid R\rangle$组$G$满足条件${\LARGEτ}'-C'(\frac{1}{2})$，任何非平凡的字在自由群中产生的$X$表示平凡元素在$G$的长度至少是最短的关系子。我们还引入了一个严格的度量条件${\LARGEτ}'_{<}$，这意味着双曲。最后，我们调查这些条件的非度量和对偶变量，并研究了小抵消理论的调和平均版本。
