Relative divergence of finitely generated groups

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.

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Keywords

divergence, relative divergence, lower distortion

Mathematical Subject Classification 2010

Primary: 20F67

Secondary: 20F65

References

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

Volume 15, issue 3 (2015)

Hung Cong Tran

Abstract