Ends of graphed equivalence relations. II (Q1001453)

For Part I see [\textit{B. D. Miller}, Isr. J. Math. 169, 375--392 (2009; Zbl 1166.03026)]. Given a graphing \(G\) of a countable Borel equivalence relation on a Polish space, the authors show that if there is a~Borel way of selecting a non-empty closed set of countably many ends from each \(G\)-component, then there is a Borel way of selecting an end or line from each \(G\)-component. Their method yields also Glimm-Effros-style dichotomies that characterize the circumstances under which: (1) there is a Borel way of selecting a point or end from each \(G\)-component; and (2) there is a Borel way of selecting a point, end or line from each \(G\)-component.