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 \((\mathcal X,\nu )\) with \(\sigma \)-finite measure \(\nu \). Let \(\Gamma \subseteq G\) be an almost normal subgroup with fundamental domain \(F\subseteq {\mathcal X}\) of finite measure. Let \(\mathcal R_G\) be the countable measurable equivalence relation on \(\mathcal X\) determined by the orbits of \(G\). Let \(\mathcal R_G| F\) 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 \(\mathcal R_G|_F\). The generators of the relation \(\mathcal R_G|_F\) are a set of transformations of the quotient space \(F\cong {\mathcal X}/ \Gamma \), in a one to one correspondence with the cosets of \(\Gamma \) in \(G\). We prove that the composition formula for these transformations is an averaged version, with coefficients in \(L^\infty (F,\nu )\), of the Hecke algebra product formula. In the situation \(G = \mathop {\mathrm{PGL}}_2(\mathbb Z[\frac{1}{p}])\), \(\Gamma =\mathop {\mathrm{PSL}}\nolimits _2(\mathbb Z)\), \(p\ge 3\) prime number, the relation \(\mathcal R_G|_F\) 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 \(\mathcal R_G|_F\) to find a canonical treeing.