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.
Hisashi AOI. Takehiko YAMANOUCHI. "Hecke pairs of ergodic discrete measured equivalence relations and the Schlichting completion." J. Math. Soc. Japan 69 (4) 1611 - 1665, October, 2017. https://doi.org/10.2969/jmsj/06941611