A note on acylindrical hyperbolicity of Mapping Class Groups

TL;DR

This paper provides a simplified proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, emphasizing the hyperbolicity of their curve graphs and the loxodromic action of pseudo-Anosov elements.

Contribution

It offers the simplest proof that these groups are acylindrically hyperbolic, highlighting the hyperbolic nature of their associated curve graphs and pseudo-Anosov dynamics.

Findings

Curve graphs are hyperbolic

Pseudo-Anosovs act as loxodromic WPDs

Mapping Class Groups are acylindrically hyperbolic

Abstract

The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on them as loxodromic WPDs.