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

Abstract

We construct explicit affine uniformly Lipschitz actions on ${\ell }^1$ 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 ${\ell }^1$-metric quasi-isometric (respectively, almost quasi-isometric, up to a logarithmic factor) to the word metric. This allows us to construct proper continuous affine uniformly Lipschitz actions on $L^1$, with undistorted orbits, for all simple groups of real rank one.

arXiv comment38 pages