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2012 Statistical hyperbolicity in groups
Moon Duchin, Samuel Lelièvre, Christopher Mooney
Algebr. Geom. Topol. 12(1): 1-18 (2012). DOI: 10.2140/agt.2012.12.1


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.


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Moon Duchin. Samuel Lelièvre. Christopher Mooney. "Statistical hyperbolicity in groups." Algebr. Geom. Topol. 12 (1) 1 - 18, 2012.


Received: 15 March 2011; Revised: 6 September 2011; Accepted: 12 October 2011; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1282.20048
MathSciNet: MR2889542
Digital Object Identifier: 10.2140/agt.2012.12.1

Primary: 20F65
Secondary: 11H06 , 52A40 , 57S30

Keywords: convex geometry , geometric group theory

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 1 • 2012
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