  The subalgebra of the tautological ring of the moduli of curves of compact
type generated by the kappa classes is studied in all genera. Relations,
constructed via the virtual geometry of the moduli of stable quotients, are
used to obtain minimal sets of generators. Bases and Betti numbers of the kappa
rings are computed. A universality property relating the higher genus kappa
rings to the genus 0 rings is stated and proved in a sequel. The lambda_g
formula for Hodge integrals arises as the simplest consequence.
