On linear divergence in finitely generated groups

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 $m\geq 2$ 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 $δ$-hyperbolic spaces that are uniformly not a quasi-line.

💡 Research Summary

The paper investigates the divergence function of a wide variety of finitely generated groups and graphs, establishing that in all the cases considered the divergence grows at most linearly. Divergence, as defined by Gersten and Gromov, measures the minimal length of a path joining two points while staying outside a prescribed ball around a third point. For one‑ended geodesic metric spaces the divergence is either infinite (if the space has more than one end) or at least linear; the authors work throughout with one‑ended spaces. A key tool is the result of Drutu‑Mozes‑Sapir, which says that if every point lies within a bounded distance of a bi‑Lipschitz infinite geodesic, then the divergence is a quasi‑isometry invariant and is well defined up to the equivalence relation ≍ (i.e. up to multiplicative and additive constants).

The central objects of study are wreath products, permutational wreath products, halo products, Houghton groups, Baumslag‑Solitar groups, Diestel‑Leader graphs, and more generally horocyclic products of proper, geodesically complete Busemann δ‑hyperbolic spaces that are uniformly not quasi‑lines. The authors develop a unified “cursor‑and‑lamps” picture: in a wreath product H≀F the element is a pair (k,f) where f∈F records the position of a cursor and k is a finitely supported function F→H describing the state of lamps. By writing any element in an optimal form
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