Property RD and hypercontractivity for orthogonal free quantum groups

Abstract: We prove that the twisted property RD fails to hold for all non Kac type, non amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $L_p-L_2$ non-commutative Khintchine inequality for free groups. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non Kac cases.