Coarse separation and splittings in hyperbolic groups

TL;DR

This paper investigates the coarse separation properties of one-ended hyperbolic groups, establishing conditions under which they are separable by subexponentially growing subsets and analyzing their splitting behavior.

Contribution

It provides a characterization of coarse separability in hyperbolic groups based on group splittings and introduces bounds on separation profiles, with applications to graph products.

Findings

Large thickened spheres are hard to cut, with exponential size cut-sets.

Hyperbolic groups not splitting over a virtually cyclic subgroup are not coarsely separable by subexponential subsets.

Polynomial lower bounds on separation profiles for certain hyperbolic groups.

Abstract

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.