Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of L 1

We define the notion of almost invariant conditionally negative definite kernel and use it to give a characterisation of groups admitting a proper uniformly Lipschitz affine action on a subspace of an L 1 space. We show that this condition is satisfied by groups acting properly on products of quasi-trees, weakly amenable groups with Cowling–Haagerup constant 1, and a-TTT-menable groups.