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Amenability of Groups and G-Sets 1 Amenability of Groups and G-Sets

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Amenability of Groups and G-Sets 1 Amenability of Groups and G-Sets Amenability of groups and G-sets 1 Amenability of groups and G-sets Laurent Bartholdi1 Abstract. This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of Gottingen¨ and the Ecole´ Normale Superieure.´ The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collabo- ration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups. 1 Introduction In 1929, John von Neumann introduced in [103] the notion of amenability of G- spaces. Fundamentally, it considers the following property of a group G acting on a set X: The right G-set X is amenable if there exists a G-invariant mean on the power set of X, namely a function m: fsubsets of Xg ! [0;1] satisfying m(AtB) = m(A) + m(B) and m(X) = 1 and m(Ag) = m(A) for all A;B ⊆ X and all g 2 G. Amenability may be thought of as a finiteness condition, since non-empty finite G-sets are amenable with m(A) = #A=#X; it may also be thought of as a fixed-point property: on a general G-set there exists a G-invariant mean; on a compact G-set there exists a G-invariant measure, and on a convex compact G-set there exists a G-fixed point, see x6; on a G-measure space there exists a G-invariant measurable family of means on the orbits, see x6.2. Amenability may be defined for other objects such as graphs and random walks on sets. If X is a G-set and G is finitely generated, then X naturally has the structure of a graph, with one edge from x to xs for every x 2 X;s 2 S. Amenability means, in the context of graphs, that there are finite subsets of X with arbitrarily small bound- ary with respect to their size. In terms of random walks, it means that there are finite subsets with arbitrarily small connectivity between the set and its complement, and equivalently that the return probability of the random walk decreases subexponen- tially in time; see x8. The definition may also be modified in another direction: rather than considering group actions, we may consider equivalence relations, or more generally groupoids. arXiv:1705.04091v1 [math.GR] 11 May 2017 The case we concentrate on is an equivalence relation with countable leaves on a standard measure space. The orbits of a countable group acting measurably naturally give rise to such an equivalence relation. This point of view is actually very valuable: 1 Ecole´ Normale Superieure,´ Paris, e-mail: [email protected] Supported by the “@raction” grant ANR-14-ACHN-0018-01 2 L. Bartholdi quite different groups (e.g. one with free subgroups, one without) may generate the same equivalence relation; see x6.2. One of the virtues of the notion of amenability of G-sets is that there is a wealth of equivalent definitions; depending on context, one definition may be easier than another to check, and another may be more useful. In summary, the following will be shown, in the text, to be equivalent for a G-set X: • X is amenable; i.e. there is a G-invariant mean on subsets of X; • There is a G-invariant normalized positive functional in `¥(X)∗, see Corol- lary 2.25; • For every bounded functions hi on X and gi 2 G the function ∑i(1 − gi) is non- negative somewhere on X, see Theorem 2.29; • For every finite subset S ⊆ G and every e > 0 there exists a finite subset F ⊆ X with #(FS n F) < e#F, see Theorem 3.23(5); • For every finite subset S ⊆ G, every e > 0 and every p 2 [1;¥) there exists a positive function f 2 `p(X) with kfs − fk < ekfk for all s 2 S, see Theo- rem 3.23(4); • There does not exist a “paradoxical decomposition” of X, namely X = Z1 t···t Zm = Zm+1 t ··· t Zm+n = Z1g1 t ··· t Zm+ngm+n for some Zi ⊂ X and gi 2 G, see Theorem 5.14(2); • There does not exist a map f : X ý and a finite subset S ⊆ G with #f −1(x) = 2 and f(x) 2 xS for all x 2 X, see Theorem 5.14(4); • There does not exist a free action of a non-amenable group H on X with the property that for every h 2 H there is a finite subset S ⊆ G with xh 2 xS for all x 2 X, see Theorem 5.15(3); • Every convex compact set equipped with a G-equivariant map from X admits a fixed point, see Theorem 6.4; • Every compact set equipped with a G-equivariant map from X admits an invari- ant measure, see Theorem 6.7; • The isoperimetric constant (Definition 8.2) of every non-degenerate G-driven random walk on X vanishes, see Theorem 8.4(2); • The spectral radius (Definition 8.2) of every non-degenerate G-driven random walk on X is equal to 1, see Theorem 8.4(3). Amenability has been given particular attention for groups themselves, seen as G-sets under right multiplication; see the next section. We stress that many results that exclusively concern groups (e.g., the recent proofs that topological full groups are amenable) are actually proven using amenable G-sets in a fundamental manner. The reason is that a group is amenable if and only if it acts on an amenable G-set with amenable point stabilizers, see Proposition 2.26. Quotients of amenable G-sets are again amenable; but sub-G-sets of amenable G-sets need not be amenable. A stronger notion will be developed in x9, that of extensively amenable G-sets. It has the fundamental property that, if p : X Y is a G-equivariant map between G-sets, then X is extensively amenable if and only if both Y and all p−1(y) are extensively amenable, the latter for the action of the stabilizer Gy. Amenability of groups and G-sets 3 We detail slightly Reiter’s characterization of amenability given above: the space `1(G) of summable functions on G is a Banach algebra under convolution, and `1(X) is a Banach `1(G)-module. We denote by v(`1G) and v(`1X) respectively the ideal 1 1 and submodule of functions with 0 sum, and by `+(G) and `+(X) the cones of positive elements. 1 Then XG is amenable if and only if for every e > 0 and every g 2 v(` G) there 1 exists f 2 `+(X) with k f gk < ek f k, see Proposition 3.25. The quantifiers may be exchanged; we call XG laminable if for every e > 0 and 1 1 every f 2 v(` X) there exists g 2 `+(G) with k f gk < ekgk, see Theorem 8.20. It has the consequence that there exists a measure m on G such that every m-harmonic function on X is constant. In case X = GG, these definitions are equivalent, but for G-sets the properties of being amenable or Liouville are in general position. Groups with free subgroups non-virt.-soluble matrix groups Non-amenable groups word-hyperbolic 1 m “HC SVNT DRACONES” small cancellation for ;m) B(n Amenable groups p1(Sg) Frankenstein’s group (virt.) free Basilica group Topological full groups of minimal Z-actions F2 Subexponentially amenable subexponential growth polynomial growth Grigorchuk’s group Z (virt.) abelian(virt.) nilpotent(virt.) polycyclic(virt.) soluble Elementary amenable Finite Z=2 Z2 1 Fig. 1 The universe of groups 1.1 Amenability of groups John von Neumann’s purpose, in introducing amenability of G-spaces, was to under- stand better the group-theoretical nature of the Hausdorff-Banach-Tarski paradox. This paradox, due to Banach and Tarski [4] and based on Hausdorff’s work [62], states that a solid ball can be decomposed into five pieces, which when appropri- ately rotated and translated can be reassembled in two balls of same size as the 4 L. Bartholdi original one. It could have been felt as a death blow to measure theory; it is now resolved by saying that the pieces are not measurable. A group is called amenable if all non-empty G-sets are amenable; and it suffices to check that the regular G-set GG is amenable, see Corollary 2.11. Using the “paradoxical decompositions” criterion, it is easy to see that the free group F2 = ha;b ji is not amenable: we exhibit a partition F2 = G1 t ··· t Gm t H1 t ··· t Hn and elements g1;:::;gm;h1;:::;hn with F2 = G1g1 t ··· t Gmgm = H1h1 t ··· t Hnhn as follows. Set −1 −2 G1 = fwords whose reduced form ends in ag [ f1;a ;a ;:::g; −1 −1 −2 G2 = fwords whose reduced form ends in a g n fa ;a ;:::g; H1 = fwords whose reduced form ends in bg; −1 H2 = fwords whose reduced form ends in b g; then F2 = G1 t G2 t H1 t H2 = G1 t G2a = H1 t H2b. The group of rotations SO3(R) contains a free subgroup F2, and even one that acts 2 freely on the sphere S ; so its orbits are all isomorphic to F2.
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