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Relative divergence of finitely generated groups

Hung Cong Tran

Algebraic & Geometric Topology 15 (2015) 1717–1769

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.

divergence, relative divergence, lower distortion
Mathematical Subject Classification 2010
Primary: 20F67
Secondary: 20F65
Received: 30 June 2014
Revised: 14 October 2014
Accepted: 25 October 2014
Published: 19 June 2015
Hung Cong Tran
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
PO Box 413
Milwaukee, WI 53201