Thickness and relative hyperbolicity for graphs of multicurves

TL;DR

This paper classifies graphs of multicurves on surfaces as hyperbolic, relatively hyperbolic, or thick based on their geometric properties and intersection patterns with subsurfaces, extending known results to a wider class.

Contribution

It generalizes previous results by providing a unified geometric framework for understanding various graphs of multicurves on surfaces.

Findings

Graphs are classified as hyperbolic, relatively hyperbolic, or thick.

The classification depends on the set of subsurfaces intersecting all vertices.

Extends known results from specific graphs to a broader family.

Abstract

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex of the graph. This extends previously established results for the pants graph and the separating curve graph to a broad family of graphs associated to surfaces.