Ends of graphed equivalence relations,II

Given a graphing of a countable Borel equivalence relation on a Polish space, we show that if there is a Borel way of selecting a non-empty closed set of countably many ends from each -component, then there is a Borel way of selecting an end or line from each -component. Our method yields also Glimm-Effros style dichotomies which characterize the circumstances under which: (1) there is a Borel way of selecting a point or end from each -component; and (2) there is a Borel way of selecting a point, end or line from each -component. The first author was supported in part by NSF Grant DMS-0140503. The second author was supported in part by NSF VIGRE Grant DMS-0502315.