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

Abstract

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.