Characterizing hierarchically hyperbolic free by cyclic groups

We algebraically characterize free by cyclic groups that have coarse medians, and prove that this is equivalent to the a priori stronger properties of being colourable hierarchically hyperbolic groups and being quasi-isometric to CAT(0) cube complexes. Our algebraic characterization involves a condition on intersections between maximal virtually $F_n\times \mathbb{Z}$ subgroups that we call having “unbranched blocks”. We also characterize hierarchical hyperbolicity of $\mathrm{\Gamma }=F_n{⋊}_{\phi }\mathbb{Z}$ in terms of a property of completely split relative train track representatives of $\phi \in \mathrm{O}\mathrm{u}\mathrm{t}(F_n)$ that we call “excessive linearity”, a slight refinement of the rich linearity condition for relative train track maps introduced by Munro and Petyt.