Rigidity and Equivalence Relations with Infinitely Many Ends

We consider groups and equivalence relations with infinitely many ends and the problem of selecting one end in a uniform manner. In general a non-amenable equivalence relation may have infinitely many ends and yet admit a Borel function selecting from each class a single end; however, we show that in the presence of an invariant Borel probability measure, the equivalence having infinitely many ends precludes the existence of a Borel method of selecting an end from each class. By analyzing the action of a group on its space of ends, we obtain a new proof of an earlier superrigidity result due to Monod and Shalom for actions of certain product groups. Finally, we discuss applications of equivalence relations with infinitely many ends to the theory of percolation and to the abstract theory of Borel equivalence relations. In particular, we show that for equivalence relations with infinitely many ends, all the amenable normal subequivalence relations are smooth.