Volume 12, issue 1 (2012)

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Statistical hyperbolicity in groups

Moon Duchin, Samuel Lelièvre and Christopher Mooney

Algebraic & Geometric Topology 12 (2012) 1–18

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 d as the generators vary, by studying the space d with various norms.

Geometric group theory, Convex geometry
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 11H06, 57S30, 52A40
Received: 15 March 2011
Revised: 6 September 2011
Accepted: 12 October 2011
Published: 13 January 2012
Moon Duchin
Department of Mathematics
Tufts University
Medford 02155
Samuel Lelièvre
Laboratoire de mathématique d’Orsay
Université Paris-Sud (Paris 11)
91405 Orsay cedex
Christopher Mooney
Department of Mathematics
Bradley University
Peoria, IL 61625