Each ring of algebraic integers in a number field gives rise to an ample (étale) groupoid and a corresponding groupoid C*-algebra. In joint work with Xin Li, we proved a rigidity theorem for this construction, showing that the groupoid (equivalently, the associated Cartan pair of C*-algebras) completely remembers the ring. I will review this result, then present on joint work with Yosuke Kubota and Takuya Takeishi where we compute groupoid homology for our groupoids. These homology computations combined with a recent breakthrough by Xin Li lead to simplicity results for the topological full groups attached to rings of algebraic integers. Time permitting, I will also briefly mention ongoing work in which we recover number-theoretic invariants from these topological full groups.