Characterization of the Haagerup property by fibred coarse embedding into Hilbert space

We show that a finitely generated, residually finite group has the Haagerup property (Gromov's a‐T‐menability) if and only if one (or equivalently, all) of its box spaces admits a fibred coarse embedding into Hilbert space. In contrast, the box spaces of a finitely generated, residually finite hyperbolic group with property (T) do not admit a fibred coarse embedding into Hilbert space, but do admit a fibred coarse embedding into an ℓ p ‐space for some p >2.