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We verify Shalom's conjecture for the simple real-rank-one Lie group Sp(n ,1)for any n: i.e. we show that it admits a metrically proper affine action on aHilbert space whose linear part is a uniformly bounded representation. Weprovide two different proofs. Both approaches crucially use results onuniformly bounded representations by Michael Cowling. The first approach isquite abstract: it uses an automatic-properness result of Shalom and requiresalmost no computations. The second approach is explicit: we deduce theproperness of cocycles from the non-continuity of a critical case of theSobolev embedding. This work is inspired from Pierre Julg's work on theBaum--Connes conjecture for Sp(n,1).

Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation