Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

Abstract

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.

Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

“…x E, which corresponds to a Seifert piece of the JSJ decomposition of the peripheral graph manifold, along with a pair of hyperbolic spaces K v and " v that will figure into the HHS structure on . The space K v is obtained via a quasimorphism constructed using the Seifert fibered structure following ideas in forthcoming work of the fourth author with Hagen, Russell, and Spriano [29], while " v is coarsely obtained by coning off boundary components of the universal covers of the base 2orbifold of this Seifert fibered manifold. We then appeal to the flat geometry of the fibers of E to construct and study certain projection maps…”

“…For each v 2 V , we will use ideas from work-in-progress of the fourth author with Hagen, Russell, and Spriano [29] to define a map…”

“…Consequently, could only be shown to be an HHS, rather than an HHG. The ideas from [29] were crucial in this extension.…”