## Algebraic & Geometric Topology

### Relative divergence of finitely generated groups

Hung Tran

#### Abstract

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

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

Dates
Revised: 14 October 2014
Accepted: 25 October 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510840973

Digital Object Identifier
doi:10.2140/agt.2015.15.1717

Mathematical Reviews number (MathSciNet)
MR3361149

Zentralblatt MATH identifier
1371.20048

#### Citation

Tran, Hung. Relative divergence of finitely generated groups. Algebr. Geom. Topol. 15 (2015), no. 3, 1717--1769. doi:10.2140/agt.2015.15.1717. https://projecteuclid.org/euclid.agt/1510840973

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