Hyperbolicity and uniformly Lipschitz affine actions on subspaces of L1$L^1$

Abstract

AbstractWe show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of an space. We also prove that every acylindrically hyperbolic group has a uniformly Lipschitz affine action on such a space with unbounded orbits. Our main tools are the ‐bicombings on hyperbolic groups constructed by Mineyev and the characterisation of acylindrical hyperbolicity in terms of actions on quasi‐trees by Balasubramanya.