On linear divergence in finitely generated groups

Abstract

In this paper, we show that wreath products of groups have linear divergence, and we generalise the argument to permutational wreath products. We also prove that Houghton groups ${\mathcal{H}}_m$ with $m\ge 2$ and Baumslag-Solitar groups have linear divergence. We explain how to generalise the argument for wreath products so that it holds for halo products of groups whose halo is large-scale commutative. Finally, we show that wreath products of graphs and Diestel-Leader graphs have linear divergence. The argument for Diestel-Leader graphs is further generalised to horocyclic products of proper, geodesically complete, Busemann $\hat{I}´$-hyperbolic spaces that are uniformly not a quasi-line.

27 pages, 7 figures. A subsection (Subsection 6.1) about horocyclic products of hyperbolic spaces with some natural conditions and a section (Section 8) about halo products of groups whose halo is large-scale commutative were added. To appear in Geometriae Dedicata