A canonical arithmetic quotient for actions of lattices in simple groups

In this paper we produce an invariant for any ergodic, finite entropy action of a lattice in a simple Lie group on a finite measure space. The invariant is essentially an equivalence class of measurable quotients of a certain type. The quotients are essentially double coset spaces and are constructed from a Lie group, a compact subgroup of the Lie group, and a commensurability class of lattices in the Lie group.