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Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$. (arXiv:2109.12949v4 [math.GR] UPDATED)

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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.

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