On the countable, measure preserving relation induced on an 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 (\cite{FM}) 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 (\cite{BC}). In the situation $G=PG{L}_2(\mathbb{Z}[\frac{1}{p}])$, $\Gamma =PS{L}_2(\mathbb{Z})$, $p\ge 3$ 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 $F$ (\cite{Ad},\cite {Hj}). We use the coset representations of the transformations generating ${\mathcal{R}}_G{|}_F$ to find a canonical treeing (\cite{Ga}).

Extensive editorial revision of the previous version, 32 pages

References in corpus (1)

• Phase transition in the Connes-Marcolli GL2-system