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We show that the orthogonal free quantum groups are not inner amenable and weconstruct an explicit proper cocycle weakly contained in the regularrepresentation. This strengthens the result of Vaes and the second author,showing that the associated von Neumann algebras are full II_1-factors andBrannan's result showing that the orthogonal free quantum groups haveHaagerup's approximation property. We also deduce Ozawa-Popa's property strong(HH) and give a new proof of Isono's result about strong solidity.

A cocycle in the adjoint representation of the orthogonal free quantum groups