This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c : X →

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We describe a stable mixed finite element method for linear elasticity in three dimensions.

Abstract. Let R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a 'treeing' of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends. IntroductionClassically, ergodic theory has studied actions of the reals or of the integers on measure spaces. More recently, it has broadened to include actions of Lie groups, algebraic groups and discrete groups. The most recent trend [6,7,9,11] has been to consider any kind of groupoid structure, where the unit space is a measure space. This includes equivalence relations on measure spaces (the case where the groupoid is principal), for which many of the fundamental techniques were developed in [3][4][5].At a conference in Santa Barbara in the late 1970s, Alain Connes suggested the study of equivalence relations with an additional piece of data: a measurably-varying simplicial complex structure on each equivalence class (see Definition 1.7). In this paper, we take up a special case of this, where the simplicial complex is a tree (see Definition 1.5).The simplest example of such a 'treed' equivalence relation comes from considering the orbits of the action of a finitely-generated free group F acting freely on a measure space. A tree structure is then obtained on each orbit by saying that two points x and y are 'adjacent' if there is some generator ge F such that gx = y or gy = x. The resulting tree is homogeneous, i.e., has a vertex transitive automorphism group. Thus, in a sense, treed equivalence relations are a groupoid analogue of free groups. In another sense (see Example 1.6.3), treed equivalence relations are analogous to foliations by manifolds of negative curvature, in the same way as a homogeneous tree is analogous to hyperbolic space (see [8, Chapter II]).Amenability is another property of equivalence relations; it is the groupoid analogue of the property of amenability for groups. It has been studied in detail by R. Zimmer and by others. The purpose of this paper is to study how amenability and treeings interact. One immediate consequence of [2] is that any amenable equivalence relation is treeable, where each equivalence 2 S. Adams class becomes tree isomorphic to the Cayley graph of Z (with generating set {±1}), i.e., to the tree with two edges belonging to every vertex.In this paper we prove two less immediate results. First, we show, in the presence of a finite invariant measure, that if the equivalence relation is amenable, then a.e. equivalence class has a very simple tree-structure:THEOREM 5.1. Let (M,R) be an amenable equivalence space with finite R-invariant measure. Let S be a treeing of (M, R). Then, for a.e. xe M, the tree R(x) has one or two ends.We prefa...

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