Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs

TL;DR

This paper demonstrates that arc and curve graphs in topology are hyperbolic spaces with specific slim triangle constants, using unicorn paths to establish their geometric properties.

Contribution

It introduces unicorn paths in arc graphs and proves their properties, leading to new bounds on the hyperbolicity of arc and curve graphs.

Findings

Arc graphs are 7-hyperbolic.

Curve graphs are 17-hyperbolic.

Unicorn paths form 1-slim triangles and are invariant under subpaths.

Abstract

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.