Hierarchical hyperbolicity of hyperbolic-2-decomposable groups

TL;DR

This paper characterizes when a graph of hyperbolic groups with 2-ended edge groups is hierarchically hyperbolic, linking it to the absence of certain distorted cyclic subgroups and deriving geometric properties.

Contribution

It provides a precise criterion for hierarchical hyperbolicity in such groups, connecting it to the absence of specific Baumslag-Solitar group quotients and establishing new geometric results.

Findings

G is hierarchically hyperbolic iff G has no distorted infinite cyclic subgroup

G satisfies quadratic isoperimetric inequality

G has finite asymptotic dimension

Abstract

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only if G does not contain certain quotients of Baumslag-Solitar groups. As a consequence, we obtain several new results about this class, such as quadratic isoperimetric inequality and finite asymptotic dimension.