Two-page paper on the notion of cost of groups and measured equivalence
relations to appear in the "What is?" series in the Notices of the AMS.Comment: 2 page

Gaboriau, Damien

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2 pagesInternational audienceTwo-page paper on the notion of cost of groups and measured equivalence relations to appear in the "What is?" series in the Notices of the AMS

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What is Cost?Damien GaboriauTo cite this version:Damien Gaboriau. What is Cost?. Notices of the AMS, American Mathematical Society, 2010,57 (10), pp.1295-1296. <ensl-00534141>HAL Id: ensl-00534141https://hal-ens-lyon.archives-ouvertes.fr/ensl-00534141Submitted on 8 Nov 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.WHAT IS COST?Damien GaboriauOrbit equivalence theory considers dynamicalsystems from the point of view of orbit equiv-alence relations. The notion of cost is a usefulinvariant in this theory.When a countable group Γ acts on a spaceit defines an equivalence relation: “to be in thesame orbit”. We consider measure preserving ac-tions on a standard probability measure space.Cost was introduced by G. Levitt in order toquantify the amount of information needed tobuild this equivalence relation.Consider for instance the Z2-action on the cir-cle R/2πZ given by two rotations a and b, whoseangles together with 2π are rationally indepen-dent. Because of commutativity, there are manyways to check that two points are in the sameorbit by using elementary jumps x ∼ a±1(x) andx ∼ b±1(x). Indeed, the information encoded inthe data {a, b} is highly redundant. Chooses in-stead some interval Iǫ ⊂ R/2πZ of length ǫ > 0,and restricts the elementary jumps x ∼ b(x) toonly those x’s in Iǫ (and retain the a-jumps).Then Φǫ := {a, b|Iǫ} still generates the or-bit equivalence relation R of the Z2-action: thesmallest equivalence relation containing all the(x, ϕ(x)) for ϕ ∈ Φǫ and x in the domain of ϕ, isR itself.In fact, because the a-orbits are dense, eachpoint z admits some a-iterate an(z) in Iǫ, so thatthe connection between z and b(z) may be recov-ered in (2n+1)-jumps, namely n times a, followedby the restriction b|Iǫ and then n times a−1. Themeasures of the domains of a and b|Iǫ sum to1 + ǫ. This is by definition the cost of Φǫ. Andit is cheaper than the cost of {a, b}. Moreover,considering ǫ tending to 0 leads one to declare Rto have cost = 1 (a priori ≤ 1, but Cost(R) ≥ 1when the classes are infinite).More generally, consider an action α of acountable group Γ on a standard Borel space(X,µ), preserving the probability measure µ. Ex-amples of such actions are plentiful, for instanceBernoulli shifts (X,µ) = (X0, µ0)Γ (Γ acting byprecomposition on functions f : Γ → X0 pre-serves the product measure µ⊗Γ0 ), or for instancethe action by multiplication of a countable sub-group of a compact group with its Haar measure.In this measure theoretic context, all the con-structions have to be measurable and sets of mea-sure 0 are neglected. Assume that the action isfree, i.e. the only element with a fixed point setof positive measure is the identity.Consider a countable family Φ = {ϕj} of iso-morphisms ϕj : Aj → Bj between Borel subsetsAj , Bj ⊂ X whose graphs are each contained inRα, i.e. for each j, each x ∈ Aj belongs to theα-orbit of ϕj(x). The cost of Φ is the numberof generators weighted by the measure of theirdomains, that is to say Cost(Φ) =∑j µ(Aj).The cost of the action α, and equivalently ofits orbit equivalence relationRα, is defined as theinfimum of the costs over all generating Φ’s:Cost(Rα) := inf{Cost(Φ) : Φ generates Rα}.It is clear, by taking the ϕj = α(γj) associatedwith a generating set (γj) of Γ that Cost(Rα) isless than or equal to the rank of Γ, i.e. its mini-mal number of generators.There is a pedantic way of defining the rank ofa countable group: as the infimum of the mea-sures of the generating subsets, for the natu-ral measure on Γ, namely the counting-measure.When applied to Rα, this gives an interesting in-terpretation of the cost. As a subset of X ×X ,Rα is simply the union of the graphs of the mapsα(γ) : X → X , for γ ∈ Γ. There is a naturalmeasure ν on Rα defined by pushing forward µby the maps sγ : x 7→ (x, γ(x)) and gatheringtogether the various measures sγ∗µ.The cost of Rα is equivalently defined asthe infimum of the ν-measures of the subsets ofRα, that are not contained in any proper sub-equivalence relation.The Z2-action above, for which we computedthe cost, belongs to a larger class of examples.Ornstein and Weiss proved the following remark-able result: for any free action of an infinite12amenable group (for instance commutative, ornilpotent, or solvable groups), the orbit equiv-alence relation may also be defined by a singletransformation of the space ψ : X → X . Forevery γ ∈ Γ and almost every x ∈ X , there isa certain iterate n(γ, x) such that ψn(γ,x)(x) =α(γ)(x). For all these actions, the infimum in thedefinition of the cost is in fact a minimum andequals 1. This pair of properties in turn, whensatisfied for some free action, implies amenabilityof Γ.The direct product of any two infinite groupsΓ1 × Γ2, with at least one infinite-order element,is easily seen to produce only cost 1 free actions(a straightforward elaboration of the above Z2-action example). This allows one to produceplenty of actions whose cost is 1 but for whichthe infimum is not attained for any generatingΦ: just consider non-amenable direct products.Recall that containing the free group Fn on n(> 1) generators prevents a group from beingamenable.When there exists a unique path of elementaryjumps x ∼ ϕ±1(x) to connecting any two pointsin the same orbit, Φ is called a treeing. Thisnotion is useful for computing the cost of someactions, and this is one of the main results in thetheory: A treeing always realizes the cost of theequivalence relation it generates [1]. If Φ is nota treeing, it is always possible to restrict someϕ ∈ Φ to a Borel subset of its domain and never-theless continue to generate the same equivalencerelation. On the other hand, if Ψ is a treeing,then restricting any ψ ∈ Ψ to a subset of its do-main breaks some connecting path and thus stopsgeneratingRα. A treeing is minimal in this sense.And the above statement claims that one cannotexpect any other Φ to appear that will generateRα at a cheaper costAs a consequence, the cost of any free actionof the free group Fn equals exactly n. Indeed,the family of transformations associated with afree generating set of the group is a treeing. Iftwo free actions of two free groups on (X,µ)are orbit equivalent (i.e. define the same or-bit equivalence relation), then the groups musthave the same rank. The orbit equivalence re-lation remembers this rank. As another conse-quence, the cost of any free action of PSL(2,Z)is 1 + 16 . Recall that PSL(2,Z) ≃ Z/3Z ∗ Z/2Zand 1 + 16 = (1−13 ) + (1−12 ).Naive strategies to produce treeings usuallycollide with the following fundamental fact: whenan infinite countable group acts freely, there isno measurable way to pick one point in each or-bit. This is because such a set D of selectedpoints would have infinitely many pairwise dis-joint translates γ(D), all with the same measure,in a finite measure space: this is a impossible.Grushko’s theorem states that the rank of afree product equals the sum of the ranks of thefactors. Similarly, the cost of a free action of afree product Γ1 ∗ Γ2 equals the sum of the costsof the action restricted to the factors Γi.Recall Schreier’s theorem: a finitely generatedinfinite normal subgroup of a free group Fn musthave finite index. This theorem extends to thosegroups Γ whose free actions have cost > 1. More-over, Schreier’s formula (p − 1) = i(n − 1) re-lates the rank n of the ambient free group Fnto the rank p of a subgroup of finite index i.This formula admits a counterpart in cost the-ory: [Cost(Rα|A) − 1] = µ(A) [Cost(Rα) − 1](compression formula), where Rα|A is the re-striction of the equivalence relation Rα to someBorel subset A ⊂ X that meets each orbit andthat is equipped with the normalized restrictedmeasure.Indeed, the computations of cost made thus farraise several open questions. Cost seems to de-pend not on the particular free action but solelyon the group. Is this true in general (fixed priceproblem)? There is moreover a strange coin-cidence with a numerical invariant, the first ℓ2Betti number β(2)1 (Γ). Namely, it seems thatβ(2)1 (Γ)+1 = Cost(Rα), when α is a free action ofan infinite group Γ, although only the inequality≤ has been proved. In particular, is it true thatactions of infinite Kazhdan property (T) groupshave cost = 1?Orbit equivalence theory and cost are relatedto several other mathematical fields, like oper-ator algebras, percolation on graphs, geometricgroup theory, descriptive set theory, etc. Muchof the recent progress in von Neumann algebrasand orbit equivalence was the result of a success-ful cross-pollination between these fields.Further reading[1] D. Gaboriau. Couˆt des relations d’e´quivalenceet des groupes. Invent. Math., 139(1):41–98,2000. (for those who want to improve theirFrench)[2] A. S. Kechris and B. D. Miller. Topics in or-bit equivalence, volume 1852 of Lect. Notes inMath.. Springer-Verlag, Berlin, 2004.[3] D. Gaboriau. On orbit equivalence of measurepreserving actions. In Rigidity in dynamics andgeometry (Cambridge, 2000), pages 167–186.Springer, Berlin, 2002.

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