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, 1 2 by measure-preserving transformations, on an infinite measure space (X, ν) with σ- 3 finite measure ν. Let Ŵ ⊆ G be an almost normal subgroup with fundamental domain 4 F ⊆ X of finitemeasure. LetRG be the countablemeasurable equivalence relation on 5 X determined by the orbits of G. Let RG|F be its restriction to F.We find an explicit 6 presentation, by generators and relations, for the von Neumann algebra associated, 7 by the Feldman-Moore (Trans AmMath Soc 234:325–359, 1977) construction, to the 8 relation RG|F . The generators of the relation RG|F are a set of transformations of the quotient space F ∼= 9 X/Ŵ, in a one to one correspondence with the cosets of Ŵ 10 in G.We prove that the composition formula for these transformations is an averaged 11 version, with coefficients in L∞(F, ν), of the Hecke algebra product formula Bost and Connes (Selecta Math (N.S.) 1:411–457, 1995). In the situation G = PGL2(Z[ 1 p 12 ]), 13 Ŵ = PSL2(Z), p ≥ 3 prime number, the relation RG|F is the equivalence relation 14 associated to a free,measure-preserving action of a free group on (p+1)/2 generators 15 on F Adams (Ergodic Theory Dyn Syst 10:1–14, 1990), Hjorth (Ann Pure Appl Logic 16 143:87–102, 2006).We use the coset representations of the transformations generating 17 RG|F to find a canonical treeing Gaboriau (Publ Math Inst Hautes Études Sci 95:93– 18 150, 2002).