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We show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of anL  1   $L^1$    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.

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