Cartesian powers of 3-manifolds

Previous results of the authors completely determine when the n-fold self-products of two 3-dimensional lens spaces are diffeomorphic; in particular, if n is odd then the fundamental group determines the diffeomorphism type. We prove that for all other irreducible geometric 3-manifolds with trivial first Betti number, the n-fold products of such manifolds with themselves are homeomorphic for some n⩾2 if and only if the manifolds themselves are homeomorphic and obtain partial results for other cases. The proofs use an assortment of techniques from 3-dimensional topology and group theory.