Actions of acylindrically hyperbolic groups on $\ell^1$

TL;DR

This paper constructs affine Lipschitz actions on  and L spaces for groups with hyperbolic features, revealing different orbit properties for acylindrically hyperbolic groups and others.

Contribution

It introduces new affine Lipschitz actions on  and L for hyperbolic groups, showing unbounded orbits for acylindrically hyperbolic groups and proper orbits for some subclasses.

Findings

Actions have unbounded orbits for acylindrically hyperbolic groups.

Actions have proper orbits for residually finite hyperbolic groups.

Induced metrics are quasi-isometric to the word metric.

Abstract

We construct affine uniformly Lipschitz actions on $\ell^1$ and $L^1$ for certain groups with hyperbolic features. For acylindrically hyperbolic groups, our actions have unbounded orbits, while for residually finite hyperbolic groups and for mapping class groups, the actions have proper orbits, with the induced $L^1$-metric quasi-isometric (respectively, almost quasi-isometric) to the word metric.