Statistical hyperbolicity in groups

Duchin, Moon¹ ; Lelièvre, Samuel² ; Mooney, Christopher³

¹Department of Mathematics, Tufts University, Medford 02155, USA
²Laboratoire de mathématique d’Orsay, Université Paris-Sud (Paris 11), 91405 Orsay cedex, France
³Department of Mathematics, Bradley University, Peoria, IL 61625, USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 1-18

Résumé

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

Détail

DOI : 10.2140/agt.2012.12.1

Classification : 20F65, 11H06, 57S30, 52A40
Keywords: Geometric group theory, Convex geometry

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