Amenable subrelations of treeable equivalence relations and applications to anti-treeability

Abstract

We study the structure of amenable subrelations of treeable measure-class-preserving equivalence relations, applying the Radon–Nikodym topography perspective initiated in our companion paper [TTD25]. For a nowhere smooth amenable subrelation, we show that the ends selected by each class in Adams's end selection theorem are precisely those which are nonvanishing along that class. From this, we derive a number of structure results about nowhere smooth amenable subrelations of treeable equivalence relations. For example, we show that every nowhere smooth amenable subrelation is contained in an essentially unique maximal amenable subrelation. This yields an anti-treeability criterion, which we use to show that every nonamenable locally compact second countable group with noncompact amenable radical is not treeable, and that every nonamenable nonunimodular treeable group has a compact open subgroup.