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 - sumj (1/2-1/dj),
where d
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|/6
where |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.
