Hyperbolicity and Uniformly Lipschitz Affine Actions on Subspaces of L1$L Boolean and 1$

We show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of an L1$L<^>1$ 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 Q$\mathbb {Q}$-bicombings on hyperbolic groups constructed by Mineyev and the characterisation of acylindrical hyperbolicity in terms of actions on quasi-trees by Balasubramanya.