Hierarchical hyperbolicity of hyperbolic-2-decomposable groups

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.