Rigid actions of hyperbolic groups admit only commutative factors

(joint work with Tattwamasi Amrutam and Eli Glasner) Let (X,\Gamma) be a minimal equicontinuous (or more generally rigid) topological dynamical system, with a discrete countable acting group. Intermediate C^-algebras of the form C^_r(\Gamma) < \mathcal{A} < C(X) \rtimes \Gamma, can be thought of as non-commutative generalizations of \Gamma-factors X \rightarrow Y as each such factor gives rise to an intermediate algebra of the form \mathcal{A} = C(Y) \rtimes \Gamma. When the group \Gamma is Gromov hyperbolic we show that this is the only possible source of intermediate algebras. The proof relies on a delicate interplay betwee two actions: The given dynamical system (X,\Gamma) and the boundary action (Z,\Gamma).