Short hierarchically hyperbolic groups I: uncountably many coarse median structures

TL;DR

This work develops the theory of short hierarchically hyperbolic groups (short HHG) and demonstrates that modifying the hierarchical structure via blowup materials and homogeneous quasimorphisms yields uncountably many coarse median structures for a broad class of groups, including the mapping class group of a 5-punctured sphere and RAAGs on triangle- and square-free graphs. The authors introduce quasimorphisms to generate bottom-level coordinate spaces (quasilines), construct blowup materials to encode cyclic directions, and then assemble these into a combinatorial HHS that is a short HHG; they further show how to add globally and locally loxodromic directions to obtain a continuum of coarse medians. The results apply to Artin groups of large/hyperbolic type, graph manifold groups, and certain Veech-extensions, and they yield a framework to study quotients and rigidity questions in a follow-up paper. The key technical contributions include a robust Dehn-filling–style toolkit for short HHG, a corrected construction avoiding gaps in ELTAG_HHS, and a systematic avenue to generate non-equivalent coarse median structures with explicit geometric models.

Abstract

We prove that the mapping class group of a sphere with five punctures admits uncountably many coarsely equivariant coarse median structures. The same is shown for right-angled Artin groups whose defining graphs are connected, triangle- and square-free, and have at least three vertices. Remarkably, in the latter case, the coarse median structures we produce are not induced by cocompact cubulations. To obtain the above results, we develop the theory of short hierarchically hyperbolic groups (HHG), which also include Artin groups of large and hyperbolic type, graph manifold groups, and extensions of Veech groups. We develop tools to modify their hierarchical structure, including using quasimorphisms to construct quasilines that serve as coordinate spaces, and this is where the abundance of coarse median structures comes from. These techniques are of independent interest, and are used in a follow-up paper with Alessandro Sisto to study quotients of short HHG. In the process, we also clarify a proof of Hagen, Martin, and Sisto on hierarchical hyperbolicity of Artin groups of large and hyperbolic type.

Figures (7)

• Figure 1: The blowup of two adjacent vertices of $\overline{X}$.
• Figure 2: A simplex of edge-type (on the left) and a simplex of triangle-type (on the right). Their links are represented by the dashed areas.
• Figure 3: How to choose $\Theta$ and $\Psi$ when $\overline{\Delta}$ is a single vertex.
• Figure 4: How to choose $\Theta$ and $\Psi$ when $\overline{\Delta}$ is an edge. In this case one can always choose $\Psi=\emptyset$.
• Figure 5: Description of the stabilisers of links, and of the associated product regions. Here $v,w\in\overline{X}^{(0)}$ are adjacent, $e=\{v,w\}$ is the edge with endpoints $v$ and $w$, and $x\in (L_v)^{(0)}$ and $y\in (L_w)^{(0)}$. Moreover $\text{P}\operatorname{Stab}_{G}(e)\coloneq \operatorname{Stab}_{G}(v)\cap\operatorname{Stab}_{G}(w)$ is the point-wise stabiliser of $e$, while $\operatorname{Stab}_{G}(e)$ is the set-wise stabiliser. Clearly $\text{P}\operatorname{Stab}_{G}(e)$ has index at most two in $\operatorname{Stab}_{G}(e)$.

Theorems & Definitions (143)

• Theorem 1: see Theorem \ref{['thm:coarse_median']}
• Corollary 2: see Corollary \ref{['cor:coarsemedian_for_all']}
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 1.1: HHS
• Remark 1.2: Normalisation
• Definition 1.4: Consistent tuple
• Theorem 1.5: Realisation