The concept of "fibred coarse embeddability for metric spaces", was introduced by X.Chen, Q.Wang ang G.Yu, in order to detect the K-theory of maximal Roe algebras of such spaces.  In this talk, we generalize this concept to the uniformly bounded circumstance.  And we prove that a finitely generated residually finite group is uniformly bounded a-T-menable iff one of its box spaces admits a uniformly bounded fibred coarse embedding into Hilbert space. Moreover, we give an example to illustrate the differences between fibred coarse embeddability and uniformly bounded fibred coarse embeddability at K-theory level. This is based on a joint work with Dapeng Zhou.

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