We show that for a strong extension of discrete measured groupoids $1\to\mathcal{S}\to\mathcal{G}\to\mathcal{Q}\to 1$ with $L\mathcal{G}$ a finite factor, $\mathcal{Q}$ has poperty (T) if and only if the inclusion of $L\mathcal{S}$ into $L\mathcal{G}$ is corigid. In particular, this implies that $\mathcal{G}$ has property (T) if and only if $L^\infty(X)\subset L\mathcal{G}$ is corigid.
Furthermore, we give the definition of an invariant random positive definite function on a discrete group, generalizing both the notion of an Invariant Random Subgroup and a character. We use von Neumann algebras to show that all invariant random positive definite functions on groups with infinite conjugacy classes which integrate to the regular character are constant.
We also show a rigidity result for subfactors that are normalized by a representation of a lattice $\Gamma$ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of $L\Gamma$ which is normalized by the natural copy of $\Gamma$ is trivial or of finite index.
Keywords: von Neumann algebras; property (T); invariant random positive definite functions; measured groupoids; characters; rigidity
