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Behrstock, Hagen and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups admitting equivariant hierarchically hyperbolic structures. The key component of our proof is that the admissible groups introduced by Croke and Kleiner always admit equivariant hierarchically hyperbolic structures. For non-geometric graph manifolds, this is contrary to a conjecture of Behrstock, Hagen and Sisto and also contrasts with results about CAT(0) cubical structures on these groups. Perhaps surprisingly, our arguments involve the construction of suitable quasimorphisms on the Seifert pieces, in order to construct actions on quasi-lines. 

We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains. The appendix, by Jacob Russell, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended. 

Abstract A finite-dimensional CAT(0) cube complexXis equipped with several well-studied boundaries. These include theTits boundary$$\partial _TX$$ ${\partial }_{T}X$(which depends on the CAT(0) metric), theRoller boundary$${\partial _R}X$$ ${\partial }_{R}X$(which depends only on the combinatorial structure), and thesimplicial boundary$$\partial _\triangle X$$ ${\partial }_{▵}X$(which also depends only on the combinatorial structure). We use a partial order on a certain quotient of$${\partial _R}X$$ ${\partial }_{R}X$to define a simplicial Roller boundary$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$. Then, we show that$$\partial _TX$$ ${\partial }_{T}X$,$$\partial _\triangle X$$ ${\partial }_{▵}X$, and$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$are all homotopy equivalent,$$\text {Aut}(X)$$ $\text{Aut}\left(X\right)$-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.