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Supplementary results obtained after the completion of our previous paper are given together with discussing some examples. A quick review of the previous paper is also included.

It is shown that for each Hecke pair of ergodic discrete measured equivalence relations, there exists a Hecke pair of groups determined by an index cocycle associated with the given pair. We clarify that the construction of these groups can be viewed as a generalization of a notion of Schlichting completion for a Hecke pair of groups, and show that the index cocycle cited above arises from “adjusted” choice functions for the equivalence relations. We prove also that there exists a special kind of choice functions, preferable choice functions, having the property that the restriction of the corresponding index cocycle to the ergodic subrelation is minimal in the sense of Zimmer. It is then proved that the Hecke von Neumann algebra associated with the Hecke pair of groups obtained above is $\ast$-isomorphic to the Hecke von Neumann algebra associated with the Hecke pair of equivalence relations with which we start.

Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property., Comment: 31 pages, Theorem 1.4 improved

We construct measures invariant with respect to equivalence relations which are graphed by horospheric products of trees. The construction is based on using conformal systems of boundary measures on treed equivalence relations. The existence of such an invariant measure allows us to establish amenability of horospheric products of random trees.