Algebraic & Geometric Topology

Relative divergence of finitely generated groups

Hung Tran

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We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten’s notion of divergence, and lower relative divergence generalizes a definition of Cooper and Mihalik. While the lower divergence of Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro and Short can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this paper, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of CAT(0) groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups.

Article information

Algebr. Geom. Topol., Volume 15, Number 3 (2015), 1717-1769.

Received: 30 June 2014
Revised: 14 October 2014
Accepted: 25 October 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

divergence relative divergence lower distortion


Tran, Hung. Relative divergence of finitely generated groups. Algebr. Geom. Topol. 15 (2015), no. 3, 1717--1769. doi:10.2140/agt.2015.15.1717.

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