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

TL;DR

This paper demonstrates that certain curve graphs associated with surfaces and strata are hierarchically hyperbolic but not Gromov hyperbolic, providing new combinatorial models for boundary structures in Teichmüller theory.

Contribution

It establishes hierarchical hyperbolicity for non-separating curve graphs and constructs new curve graph analogues for marked strata of abelian differentials.

Findings

Non-separating curve graphs are hierarchically hyperbolic.

Constructed curve graph analogues for marked strata.

These graphs are not Gromov hyperbolic.

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.