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We define the notion of almost invariant conditionally negative definitekernel and use it to give a characterisation of groups admitting a properuniformly Lipschitz affine action on a subspace of an $L^1$ space. We show thatthis condition is satisfied by groups acting properly on products ofquasi-trees, weakly amenable groups with Cowling-Haagerup constant 1, anda-TTT-menable groups.

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