Hierarchical hyperbolicity of admissible curve graphs and the boundary of marked strata

TL;DR

The paper develops a coarse-geometric framework for framed surfaces and the boundaries of strata of abelian differentials. It defines the admissible curve graph $\\mathscr{C}_{adm}(S,\\phi)$ and proves it is hierarchically hyperbolic but not Gromov hyperbolic, linking its geometry to a Vokes-style model $\\mathcal{K}$ built from a witness system, via an intermediate genus-separating graph $\\mathcal{G}$. It then extends these ideas to boundary combinatorics of strata, introducing boundary-curve graphs $\\mathscr{C}(\\overline{\\mathcal{H}_{\\phi}})$ and divisorial models $\\mathscr{D},\\mathscr{E}$ that capture degeneration patterns through multi-scale differentials and level splittings. The authors establish quasi-isometries between the various models and the boundary graphs, using subsurface projections and transitivity results for framed mapping class groups to deduce hierarchical hyperbolicity in the boundary setting as well. In sum, the work unifies framed mapping-class geometry with strata boundary combinatorics via a robust HHS framework, providing tools to study distortions, projections, and degenerations in both the surface and stratum contexts.

Abstract

We show that for any surface of genus at least 3 equipped with any choice of framing, the graph of non-separating curves with winding number 0 with respect to the framing is hierarchically hyperbolic but not Gromov hyperbolic. We also describe how to build analogues of the curve graph for marked strata of abelian differentials that capture the combinatorics of their boundaries, analogous to how the curve graph captures the combinatorics of the augmented Teichmueller space. These curve graph analogues are also shown to be hierarchically, but not Gromov, hyperbolic.

Figures (14)

• Figure 1: GSBs and auxiliary curves as in the proof of Proposition \ref{['prop:transitiveadm']}.
• Figure 2: On the left, the subsurfaces involved in the proof of Lemma \ref{['lem:adm_bdd_dist']}. On the right, surgering an arbitrary arc $\beta'$ from $d$ to $e$ along $a \cap V_c$ to obtain a disjoint arc $\beta$.
• Figure 3: Building a genus-separating curve out of a cycle in the dual graph.
• Figure 4: Building a curve that cuts off $A$, and is hence admissible. The highlighted curves are in $A$. In this example, $A_2$ and $B_2$ are empty, so the arc from $B_1$ to $A_3$ meets $c \cap Z$ exactly once.
• Figure 5: The curves $p_1,p_2$ cobounding the pair of pants $P$. The arcs $\gamma_1$ and $\gamma_2$ cut $S \setminus P$ into "left" and "right" sides.

Theorems & Definitions (80)

• Theorem A
• Theorem B
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 2.1
• Lemma 2.2: Humphries--Johnson
• Remark 2.3
• Definition 2.4
• Theorem 2.5