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

Abstract

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

Final version, 8 pages. To appear in Bulletin of the London Mathematical Society. There was a conceptual mistake in the previous version. Section 3 has been considerably expanded in order to fix it

Cited by in corpus (2)

• Hyperbolic Metric Spaces and Stochastic Embeddings
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