In this paper, we introduce a geometric statistic called the sprawl of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products and for certain solvable groups. In free abelian groups, the word metrics are asymptotic to norms induced by convex polytopes, causing several kinds of group invariants to reduce to problems in convex geometry. We present some calculations and conjectures concerning the values taken by the sprawl statistic for the group as the generators vary, by studying the space with various norms.
Moon Duchin. Samuel Lelièvre. Christopher Mooney. "Statistical hyperbolicity in groups." Algebr. Geom. Topol. 12 (1) 1 - 18, 2012. https://doi.org/10.2140/agt.2012.12.1