A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces

TL;DR

The paper proves a true converse to the combinatorial HHS criterion: a hierarchically hyperbolic space with natural extra hypotheses (weak wedges, clean containers, orthogonals for non-split domains, and dense product regions) is quasi-isometric to a combinatorial HHS. It provides an explicit construction of a CHHS, via a blow-up of the minimal orthogonality graph and a realisation map, and shows the resulting structure is equivariant under group actions, yielding CHHS-based hierarchical groups. The framework also clarifies how CHHS structure arises in canonical examples like mapping class groups, and it elucidates connections to lattice-theoretic concepts in the nesting/orthogonality poset. Additionally, the paper discusses necessary hypotheses by presenting a counterexample where dropping the non-split orthogonality condition obstructs the construction, and it applies the theory to derive two CHHS structures for mapping class groups. Overall, the results provide a blueprint for identifying, constructing, and classifying HHSs via combinatorial tools, with implications for new HHG examples and deeper structural insights.

Abstract

The combinatorial hierarchical hyperbolicity criterion is a very useful way of constructing new hierarchically hyperbolic spaces (HHSs). We show that, conversely, HHSs satisfying natural assumptions (satisfied, for example, by mapping class groups) admit a combinatorial HHS structure. This can be useful in constructions of new HHSs, and also our construction clarifies how to apply the combinatorial HHS criterion to suspected examples. We also uncover connections between HHS notions and lattice theory notions.

Figures (10)

• Figure 1: $U$ and $V$ are orthogonal, $\sqsubseteq$-minimal domains. Therefore, the cone over $({\mathcal{C}} U)^{(0)}$ (here, in red) and the cone over $({\mathcal{C}} V)^{(0)}$ (in blue) form a join. Notice that any two points in $({\mathcal{C}} U)^{(0)}$ are not adjacent.
• Figure 2: All cases in which $\mathrm{Lk}(\Delta)$ (represented by the dashed ellipses) is not a non-trivial join.
• Figure 3: Schematic representation of the simplex $\Pi$. Each cell describes how $\Pi_U$ is defined whenever the domain $U$ belongs to the area given by the intersection between the row label and the column label (for example, if $U\in\bar{\Sigma}\cap \mathrm{Lk}(\bar{\Delta})$ we have that $\Pi_U=\Sigma_U$).
• Figure 4: The simplices involved in the construction of Lemma \ref{['lem:edges_in_link']}, where edges denote joins in $\bar{X}$. Actually, $\bar{\Sigma}$ and $\bar{\Delta}\star\bar{\Phi}$ need not be disjoint, but none of them can contain $V$ or $W$.
• Figure 5: The simplices of $\bar{X}$ involved in the second case of the proof of Lemma \ref{['lem:edges_in_link']}, where edges represent joins. By construction $\bar{\sigma}_v=\bar{\Sigma}\star\bar{\Psi}_v\star\bar{\Theta}_v$, and similarly for $\bar{\sigma}_w=\bar{\Sigma}\star\bar{\Psi}_w\star\bar{\Theta}_w$.

Theorems & Definitions (155)

• Theorem 1: see Theorem \ref{['thm:main']}
• Theorem 2
• Theorem 3
• Remark 1: Relation with clean markings
• Definition 1.1: HHS
• Lemma 1.3: DHS
• Remark 1.4: Normalisation
• Definition 1.6: Consistent tuple
• Theorem 1.7: Realisation HHS_II
• Definition 1.8: Product regions and factors