# From Hierarchical to Relative Hyperbolicity

## TL;DR

This paper introduces a new combinatorial criterion to identify when hierarchically hyperbolic spaces are relatively hyperbolic, with applications to surface graphs and Teichmüller space.

## Contribution

It provides a novel, combinatorial characterization of relative hyperbolicity within hierarchically hyperbolic spaces and groups, extending understanding of their geometric structures.

## Key findings

- Separating curve graph of a surface with zero or two punctures is relatively hyperbolic.
- The criteria recover Brock and Masur's theorem on Weil-Petersson metric hyperbolicity.
- New formulation of relative hyperbolicity in terms of hierarchy structures.

## Abstract

We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmuller space for surfaces with complexity three.