#### Volume 15, issue 3 (2015)

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Relative divergence of finitely generated groups

### Hung Cong Tran

Algebraic & Geometric Topology 15 (2015) 1717–1769
##### 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 $CAT\left(0\right)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups.

##### Keywords
divergence, relative divergence, lower distortion
Primary: 20F67
Secondary: 20F65
##### Publication
Received: 30 June 2014
Revised: 14 October 2014
Accepted: 25 October 2014
Published: 19 June 2015
##### Authors
 Hung Cong Tran Department of Mathematical Sciences University of Wisconsin–Milwaukee PO Box 413 Milwaukee, WI 53201 USA http://pantherfile.uwm.edu/hctran/Hung.html