CAT(-1) metrics on small cancellation groups

TL;DR

This paper proves that groups satisfying certain small cancellation conditions act geometrically on CAT(-1) spaces, showing that random groups at low density are also CAT(-1), through a direct hyperbolic structure construction.

Contribution

It provides a new geometric construction demonstrating that small cancellation groups under specific conditions are CAT(-1), extending understanding of their geometric properties.

Findings

Groups with uniform C'(1/6) satisfy CAT(-1) geometry

Random groups at density <1/12 are CAT(-1)

Constructs a hyperbolic structure on the presentation complex

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.