On the Countable Measure-Preserving Relation Induced on a Homogeneous Quotient by the Action of a Discrete Group

We consider a countable discrete group G acting ergodicaly and a.e. freely, by measure-preserving transformations, on an infinite measure space ( X , ν ) with σ -finite measure ν . Let Γ ⊆ G be an almost normal subgroup with fundamental domain F ⊆ X of finite measure. Let R G be the countable measurable equivalence relation on X determined by the orbits of G . Let R G | F be its restriction to F . We find an explicit presentation, by generators and relations, for the von Neumann algebra associated, by the Feldman-Moore construction, to the relation R G | F . The generators of the relation R G | F are a set of transformations of the quotient space F ≅ X / Γ , in a one to one correspondence with the cosets of Γ in G . We prove that the composition formula for these transformations is an averaged version, with coefficients in L ∞ ( F , ν ) , of the Hecke algebra product formula. In the situation G = PGL 2 ( Z [ 1 p ] ) , Γ = PSL 2 ( Z ) , p ≥ 3 prime number, the relation R G | F is the equivalence relation associated to a free, measure-preserving action of a free group on ( p + 1 ) / 2 generators on F . We use the coset representations of the transformations generating R G | F to find a canonical treeing.