# Gromov's article on Hyperbolic groups

Here is the part of Gromov's paper" "Hyperbolic groups, Math Sci Res. inst Pub B. Springer, 1987" where**combinatorial curvature**appears first: this is a graph theoretical notion and up to normalization equal to

K = 1 - sumwhere d_{j}(1/2-1/d_{j}),

_{j}are the cardinalities of the neighboring face degrees. If the graph is two dimensional implying that d

_{j}=3, this simplifies to

K = 1-|S|/6where |S| is the cardinality of the sphere of radius 1. The last formula shows more clearly the "so called" 1/5-condition for nonpositive curvature and that if |S|>6, then the curvature of the graph is strictly negative.