1607.02580Jul 9, 201612 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
CAT(-1) metrics on small cancellation groups
Samuel Brown
Abstract
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.