Strong 1-boundedness of unimodular orthogonal free quantum groups

Abstract

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors [Formula: see text] have Jung’s strong [Formula: see text]-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in [Formula: see text] dimensions [Formula: see text]. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in [Formula: see text]-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a [Formula: see text]-bounded set without losing [Formula: see text]-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong [Formula: see text]-boundedness.