Behaviour of entropy under bounded and integrable orbit equivalence
aa r X i v : . [ m a t h . D S ] N ov BEHAVIOUR OF ENTROPY UNDER BOUNDED ANDINTEGRABLE ORBIT EQUIVALENCE T IM A USTIN
Abstract
Let G and H be infinite finitely generated amenable groups. This paperstudies two notions of equivalence between actions of such groups on stan-dard Borel probability spaces. They are defined as stable orbit equivalencesin which the associated cocycles satisfy certain tail bounds. In ‘integrablestable orbit equivalence’, the length in H of the cocycle-image of an elementof G must have finite integral over its domain (a subset of the G -system),and similarly for the reverse cocycle. In ‘bounded stable orbit equivalence’,these functions must be essentially bounded in terms of the length in G . ‘In-tegrable’ stable orbit equivalence arises naturally in the study of integrablemeasure equivalence of groups themselves, as introduced recently by Bader,Furman and Sauer.The main result is a formula relating the Kolmogorov–Sinai entropies oftwo actions which are equivalent in one of these ways. Under either of thesetail assumptions, the entropies stand in a proportion given by the compres-sion constant of the stable orbit equivalence. In particular, in the case of fullorbit equivalence subject to such a tail bound, entropy is an invariant. Thiscontrasts with the case of unrestricted orbit equivalence, under which all freeergodic actions of countable amenable groups are equivalent. The proof usesan entropy-bound based on graphings for orbit equivalence relations, and inparticular on a new notion of cost which is weighted by the word lengths ofgroup elements. Contents Some preliminaries on stable orbit equivalence and cocycles 144 Finite-index subgroups 205 Virtually Euclidean groups 256 Proof of the entropy formula using derandomization 307 Subrelations, graphings and a new notion of cost 338 Constructing low-cost graphings 379 Proof of the derandomization results 4710 Further questions 49
Let G and H be finitely generated discrete groups, and let T : G ñ p X, µ q and S : H ñ p Y, ν q be free ergodic actions on standard Borel probability spaces. Atriple such as p X, µ, T q is called a G -system , and similarly for H . Let |¨| G and |¨| H be length functions on the two groups given by some choice of finite symmetricgenerating sets, and let d G and d H be the associated right-invariant word metrics.The generating sets may be written as the unit balls B G p e G , q and B H p e H , q inthese metrics.Recall that a stable orbit equivalence (or SOE ) between p X, µ, T q and p Y, ν, S q consists of (i) measurable subsets U Ď X and V Ď Y of positive measure, and(ii) a bi-measurable bijection Φ : U ÝÑ V which satisfies µ p Φ ´ A q µ p U q “ ν p A q ν p V q for all measurable A Ď V and Φ p T G p x q X U q “ S H p Φ p x qq X V for µ -a.e. x. If µ p U q “ ν p V q “ , then Φ is simply an orbit equivalence , and the systemsare said to be orbit equivalent . We often indicate a stable orbit equivalence by Φ : p X, µ, T q p Y, ν, S q .A stable orbit equivalence can be described in terms of a pair of maps whichconvert the G -action on the domain to the H -action on the target and vice-versa.2or this purpose we make the following definition. An H -valued partial cocycleover p X, µ, T q is a pair p α, U q in which U Ď X is measurable and α : tp g, x q P G ˆ X : x P U X T g ´ U u ÝÑ H is a measurable function which satisfies the cocycle identity: α p gk, x q “ α p g, T k x q α p k, x q whenever g, k P G and x P U X T k ´ U X T p gk q ´ U. If Φ : p X, µ, T q p Y, ν, S q is a stable orbit equivalence, and U and V arerespectively the domain and image of Φ , then Φ may be described in terms ofan H -valued partial cocycle p α, U q over p X, µ, T q and a G -valued partial cocycle p β, V q over p Y, ν, S q . They are defined by requiring that Φ p T g x q “ S α p g,x q p Φ p x qq whenever x P U X T g ´ U and Φ ´ p S h y q “ T β p h,y q p Φ ´ p y qq whenever y P V X S h ´ V. These equations specify the cocycles uniquely because the actions are free. Com-paring these equations gives the following relations of inversion between α and β : β p α p g, x q , Φ p x qq “ g and α p β p h, y q , Φ ´ p y qq “ h. (1)The category of probability-preserving actions and orbit equivalences has along history in ergodic theory. If G and H are amenable then the resulting equiv-alence relation on systems turns out to be trivial: all free ergodic actions of count-able amenable groups are orbit equivalent. This is the Connes–Feldman–Weissgeneralization of Dye’s theorem: see [Dye59, Dye63, CFW81]. On the otherhand, if G is amenable and p X, µ, T q is a free ergodic G -action, then a free er-godic action of another group H can be orbit equivalent to p X, µ, T q only if H isalso amenable: see, for instance, [Zim84, Section 4.3]. Among actions of non-amenable groups the relation of orbit equivalence is more complicated.The generalization to stable orbit equivalence has become important becauseof its relationship with measure equivalence of groups. For any countable groups G and H , a measure coupling of G and H is a σ -finite standard Borel mea-sure space p Ω , m q together with commuting m -preserving actions G, H ñ Ω which both have finite-measure fundamental domains. If a measure coupling ex-ists then G and H are measure equivalent . This notion was introduced by Gro-mov in [Gro93, Subsection 0.5.E] as a measure-theoretic analog of quasi-isometry.3f p Ω , m q is a measure coupling of G and H , then one can use fundamentaldomains for the G - and H -actions to produce finite-measure-preserving systemsfor G and H that are stably orbit equivalent. On the other hand, given a stable or-bit equivalence between a G -system and an H -system, they can be reconstructedinto a measure coupling of the groups: see [Fur99, Theorem 3.3], where Furmangives the credit for this result to Gromov and Zimmer. On account of this corre-spondence, one can also describe a measure coupling in terms of cocycles overthose finite-measure-preserving systems. This time one obtains cocycles over thewhole systems, not just partial cocycles. In general, it is fairly easy (though notcanonical) to extend a partial cocycle to a whole system (this is well-known, butsee Proposition 3.2 below for a careful proof).By the aforementioned result of Zimmer, if G is amenable then H can bemeasure equivalent to G only if H is also amenable. On the other hand, anyamenable group does have actions which are free and ergodic, such as the non-trivial Bernoulli shifts, so the theorem of Connes, Feldman and Weiss shows thatany two amenable groups are measure equivalent. Recent work of Bader, Furman and Sauer [BFS13] has introduced a refinementof measure equivalence called ‘integrable measure equivalence’. It is obtainedby imposing an integrability condition on the cocycles α and β that appear inthe description of a measure coupling. Their original results are for hyperbolicgroups, but recently this notion has also been studied for amenable groups. Itseems to be a significantly finer relation than measure equivalence. The growthtype of the groups is an invariant, and among groups of polynomial growth thebi-Lipschitz type of the asymptotic cone is an invariant: both of these results areproved in [Aus16].The present paper studies stable orbit equivalences which are subject to similarconditions on the integrability or boundedness of their cocycles. It may be seen asan ergodic theoretic counterpart to the study of integrable measure equivalence,or as a continuation of the study of ‘restricted orbit equivalences’ within ergodictheory.Because stable orbit equivalences are described in terms of partial cocycles,we must be a little careful in the choice of integrability condition to impose. Thispaper focuses on two alternatives. Let Φ : p X, µ, T q p Y, ν, S q be an SOE andlet p α, U q and p β, U q be the partial cocycles which describe it. • We say that Φ is a bounded stable orbit equivalence , or SOE , if there is4 finite constant C such that | α p g, x q| H ď C | g | G for µ -a.e. x P U X T g ´ U and | β p h, y q| G ď C | h | H for ν -a.e. y P V X S h ´ V for all g P G and h P H (regarding this condition as vacuous if U X T g ´ U or V X S h ´ V has measure zero). • We say that Φ is an integrable semi-stable orbit equivalence , or SSOE ,if p α, U q may be extended to a full cocycle σ : G ˆ X ÝÑ H which satisfiesthe integrability condition ż X | σ p g, x q| H µ p dx q ă 8 @ g P G, and similarly for p β, V q . Beware that the extensions of p α, U q and p β, V q are not required to satisfy any extended version of (1) beyond their originaldomains.We use the term ‘semi-stable’ for the second possibility because it requiresthat α have an extension to all of G ˆ X which is integrable; it depends on morethan just the values taken by α itself. We would call Φ an integrable stable orbitequivalence or SOE if we required only that ż U X T g ´ U | α p g, x q| H µ p dx q ă 8 @ g P G. This is formally weaker than both SSOE and SOE . The main result of thispaper, Theorem A below, concerns SOE and SSOE , but I do not know whetherit holds also for SOE .We write OE and OE for the special cases of the above notions when dom Φ and img Φ both have full measure.In the setting of single probability-preserving transformations, a classical re-sult of Belinskaya [Bel68] asserts that two transformations S and T are integrablyorbit equivalent if and only if S is isomorphic to either T or T ´ . Later, sev-eral works studied other notions of ‘restricted’ orbit equivalence for probability-preserving transformations, motived by Kakutani equivalence and Feldman’s in-troduction of loose Bernoullicity: see for instance [ORW82] and [Rud85]. Manyof those ideas have been generalized to actions of Z d for d ě and then to more5eneral amenable groups, culminating in the very abstract formulation of Kam-meyer and Rudolph in [KR97, KR02]. For Z d -actions with d ě , Fieldsteeland Friedman [FF86] have shown that several natural properties are not invari-ant under integrable, or even bounded, OE, including discrete spectrum, mixing,and the K property. However, entropy is an invariant. Indeed, it is fairly easy toshow that OE for Z d -systems implies Kakutani equivalence in the sense devel-oped in [Kat77, dJR84] (see Section 5 below), and those works include the resultthat entropy is invariant under Kakutani equivalence.The present work extends this last conclusion to SOE and SSOE and togeneral discrete amenable groups. Theorem A
Suppose that G and H are amenable, that p X, µ, T q and p Y, ν, S q are as above, and that Φ : U ÝÑ V is either a SOE or a SSOE . Then µ p U q ´ h p µ, T q “ ν p V q ´ h p ν, S q . Remark . It suffices to assume that only one of G and H is amenable, since theexistence of the stable orbit equivalence then implies that the other is too. ⊳ We prove Theorem A in two parts.The first part handles the case of the Euclidean lattices Z d . In this case, ourvarious notions of stable orbit equivalence turn out to imply Kakutani equivalence,one of the more classical notions of restricted orbit equivalence. Theorem B
Suppose that p X, µ, T q is a Z d -system and p Y, ν, S q is a Z D -system.If they are SOE then they are SSOE , and if they are SSOE then d “ D and theyare Kakutani equivalent. This will be proved in Section 5. Theorem A follows for these groups be-cause it is known how entropy transforms under Kakutani equivalences of Z d -systems [dJR84].Moreover, a fairly standard construction (see Section 4) allows one to passbetween groups and their finite-index subgroups, and so from Theorem B we candeduce Theorem A for all finitely generated, virtually Abelian groups.6n the second part of the proof, all remaining cases are deduced from a resultthat we call ‘OE derandomization’. It asserts that if a SOE between two systems isa SSOE or SOE , then it is lifted from a SOE between two factor systems havingarbitrarily low entropy (that is, having ‘arbitrarily little randomness’). Curiously,this result seems to require super -linear growth of the acting groups, and so itcannot be used to prove Theorem A for virtually cyclic groups. Thus we needboth approaches to prove Theorem A in general.A simpler version of derandomization can be observed among arbitrary inte-grable cocycles from a system to a group. We state (and later prove) this resultfirst, as motivation for the orbit-equivalence result that we need. Theorem C (Cocycle derandomization)
Let G be amenable and have super-linear growth, let p X, µ, T q be a G -system, and let σ : G ˆ X ÝÑ H be anintegrable cocycle over T . For any ε ą there is a cocycle τ cohomologous to σ over T such that the factor of p X, µ, T q generated by τ has entropy less than ε . For general cocycles G ˆ X ÝÑ H , not necessarily arising from an SOE,boundedness and integrability are defined in Subsection 2.1. For cocycles whichare not partial, boundedness implies integrability, so Theorem C applies in partic-ular to all bounded cocycles.Now suppose that Φ : p X, µ, T q p Y, ν, S q is an SOE. If U Ď dom Φ is measurable and has positive measure, then the restriction Φ | U still defines anSOE, different from Φ in that the domain and image have been made smaller. The restriction of Φ to U is always understood as a SOE in this way. Theorem D (Orbit-equivalence derandomization ) Let G be amenable andhave super-linear growth, and let Φ : p X, µ, T q p Y, ν, S q be an SOE whichis either a SSOE or a SOE . Let ε ą . Then there are • a measurable subset U Ď dom Φ with µ p U q ą , • factor maps π : p X, µ, T q ÝÑ p X , µ , T q and ξ : p Y, ν, S q ÝÑ p Y , ν , S q whose target systems are still free, • and a SOE Φ : p X , µ , T q p Y , ν , S q such that1. h p µ , T q ă ε and2. the following diagram commutes: X, µ, T q / / Φ | U / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , T q / / Φ / / p Y , ν , S q . For groups of super-linear growth, Theorem A is deduced from Theorem Din Section 6. Then Sections 7 and 8 develop some more technical results, beforeTheorems C and D are proved in Section 9. Those technical results include a newnotion of cost for a graphing on a Borel orbit equivalence relation which takes intoaccount the word lengths of different group elements, and may be of independentinterest. It appears in Definition 7.4.
Acknowledgements
This paper emerged from an ongoing collaboration with Uri Bader, Lewis Bowen,Alex Furman and Roman Sauer. I am also grateful to Oded Regev, DamienGaboriau and Brandon Seward for some useful references. Finally, I thank theanonymous reviewer for suggestions which clarified various technical steps in theproofs.
All measure spaces in this paper are standard Borel and σ -finite. Most are proba-bility spaces. Measure spaces are denoted by pairs such as p X, µ q ; the σ -algebraof this space will be denoted by B X when it is needed.An observable on a measure space p X, µ q is a measurable function ϕ from X to a countable set, and a partial observable on p X, µ q is a pair p ϕ, U q consistingof a measurable subset U Ď X and a measurable function ϕ from U to a countableset. A G -system is a triple p X, µ, T q consisting of a standard Borel probabilityspace p X, µ q and a µ -preserving measurable action T of G on that space. It is free if the orbit-map g ÞÑ T g x is injective for µ -a.e. x . The Borel orbit equivalence re-lation of this action is denoted by R T . We assume standard definitions and resultsabout orbit equivalence and cocycles over such systems: see, for instance, [Zim84,Section 4.2]. 8onventions seem a little less settled in relation to stable orbit equivalence, andI do not know of a canonical reference. It appears most often in connection withmeasure equivalence of groups, such as in [Fur99, Fur11, Gab02, Gab05, Sha04].The present paper uses slightly different conventions, since our interest is in thesystems and not just the groups. But I have followed [Fur99, Section 2] wherepossible.If H is another discrete group, then an H -valued partial cocycle over p X, µ, T q is a pair p α, U q in which U Ď X is measurable and α : tp g, x q : x P U X T g ´ U u ÝÑ H is a measurable function satisfying the cocycle identity α p gk, x q “ α p g, T k x q α p k, x q whenever g, k P G and x P U X T k ´ U X T p gk q ´ U. This reduces to the usual notion of a cocycle if U “ X . We sometimes write α g for the function α p g, ¨q : U X T g ´ U ÝÑ H, and if x P U then we write α x for the function α p¨ , x q : t g P G : T g x P U u ÝÑ H. A partial cocycle p α, U q is non-trivial if µ p U q ą . Two partial cocycles areconsidered equal if their sets are equal modulo µ and their functions agree µ -a.e.If p α, U q is a partial cocycle over p X, µ, T q and V Ď U is measurable, thenthe restriction of p α, U q to V is the partial cocycle p α | V , V q where α | V is therestriction of the map α to the set tp g, x q : x P V X T g ´ V u . To lighten notationwe sometimes write this restriction as p α, V q .A cocycle α or partial cocycle p α, U q is bounded if there is a finite constant C such that | α p g, x q| H ď C | g | G for µ -a.e. x P U X T g ´ U, for all g P G. A cocycle α (not partial) is integrable if ż | α p g, x q| H µ p dx q ă 8 @ g P G. Clearly a bounded cocycle is integrable. These usages are consistent with thedefinitions of SOE and SSOE in the Introduction.9f α : G ˆ X ÝÑ H is a cocycle, then it is bounded if and only if each of thefinitely many functions | α p s, ¨q| H , s P B G p e G , q , is essentially bounded on X . The forward implication here is immediate, and thereverse follows by writing a general element of G as g “ s ℓ ¨ ¨ ¨ s with ℓ “ | g | G and s , . . . , s ℓ P B G p e G , q , and then using the cocycle identity α p g, x q “ α p s ℓ , T s ℓ ´ ¨¨¨ s x q ¨ ¨ ¨ α p s , x q . (2)However, we cannot argue this way for a partial cocycle p α, U q , since the factorson the right-hand side of (2) may not all be defined for arbitrary x P U X T g ´ U .This is why we use the definition of boundedness given above. Let p X, µ q be a probability space. If µ is atomic, then its Shannon entropy is H p µ q : “ ´ ÿ x P X µ t x u log µ t x u , with the usual interpretation “ .If ϕ : X ÝÑ A is an observable, then its Shannon entropy is H µ p ϕ q : “ H p ϕ ˚ µ q . If U Ď X is measurable, then its Shannon entropy is defined to be that of theindicator function U : more explicitly, H µ p U q : “ ´ µ p U q log µ p U q ´ µ p X z U q log µ p X z U q . If p X, µ q is a probability space and p ϕ, U q is a partial observable on it, thenthe Shannon entropy of p ϕ, U q is defined to be H µ p ϕ ; U q : “ H µ p U q ` µ p U q ¨ H µ | U p ϕ q , where µ | U is the measure µ conditioned on U : that is, µ | U p V q : “ µ p V X U q{ µ p U q . µ p U q “ , then we set H µ p ϕ ; U q “ by convention. If p ϕ, U q is a partialobservable and V Ď U is measurable, then we abbreviate H µ p ϕ | V ; V q to just H µ p ϕ ; V q .Observe that, if p ϕ, U q is a partial observable and ˚ is an abstract point outsidethe range of ϕ , then we can define a new observable ϕ ˚ by ϕ ˚ p x q “ " ϕ p x q if x P U ˚ if x P X z U, and we obtain H µ p ϕ ; U q “ H µ p ϕ ˚ q .Now let G be a discrete amenable group with a Følner sequence p F n q n ě , andlet p X, µ, T q be a G -system. If ϕ : X ÝÑ A is an observable and F Ď G is finite,let ϕ F : “ p ϕ ˝ T g q g P F : X ÝÑ A F . The factor generated by ϕ is the σ -algebra of subsets of X generated by the level-sets of ϕ and all their images under T g , g P G . If p ϕ, U q is a partial observable,then the factor it generates is defined to be the factor generated by ϕ ˚ , the newobservable constructed above.As is standard, the Kolmogorov–Sinai (‘ KS ’) entropy of the system p X, µ, T q and observable ϕ is h p µ, T, ϕ q : “ lim n ÝÑ8 | F n | H µ p ϕ F n q . This may be calculated using any Følner sequence for G . Then the KS entropyof p X, µ, T q is the supremum of h p µ, T, ϕ q over all observables ϕ . It is denotedby h p µ, T q . By the Kolmogorov–Sinai theorem, the quantity h p µ, T, ϕ q is alwaysequal to the KS entropy of the factor of p X, µ, T q generated by ϕ .The subadditivity of Shannon entropy has the immediate consequence h p µ, T, ϕ q ď H µ p ϕ q . We extend this to a partial observable p ϕ, U q by defining h p µ, T, p ϕ, U qq to bethe KS entropy of the factor generated by p ϕ, U q . By writing this in terms of theextended observable ϕ ˚ , we immediately obtain also h p µ, T, p ϕ, U qq ď H µ p ϕ ; U q . (3)The following useful estimate may be well-known, but I have not found areference. It was shown to me by Alex Furman.11 emma 2.1. Let |¨| G be a length function on G corresponding to a finite symmetricgenerating set. For every ε ą there exists C ε ă 8 such that the following holds.If p g is a value in r , s for every g P G zt e G u , then ÿ g ‰ e G r´ p g log p g ´ p ´ p g q log p ´ p g q ‰ ď C ε ÿ g ‰ e G | g | G p g ` ε. In particular, if p X, µ q is a probability space and α : X ÝÑ G zt e G u is an observ-able, then H µ p α q ď C ε ż | α p x q| G µ p dx q ` ε Proof.
First, Markov’s Inequality gives |t g P G zt e G u : p g ě { u| ď ÿ g ‰ e G p g . On the other hand, if p g ď { then ´p ´ p g q log p ´ p g q ď ´ p g log p g . We may therefore bound the desired sum as follows: ÿ g ‰ e G r´ p g log p g ´ p ´ p g q log p ´ p g q ‰ ď ÿ g ‰ e G p´ p g log p g q ` log 2 ¨ |t g P G zt e G u : p g ě { u|ď ÿ g ‰ e G p´ p g log p g q ` ÿ g ‰ e G | g | G p g . It therefore suffices to prove that ř g ‰ e G p´ p g log p g q may be bounded in terms of ř g ‰ e G | g | G p g in the desired way.Next, since G is finitely generated, there is a finite constant c such that | B G p e G , n q| ď c n @ n ě . For each n ě , let q n : “ ÿ | g | G “ n p g . µ g : “ p g { q | g | G for all g P G , interpreting this as if q | g | G “ . Pro-vided q n ‰ , the tuple p µ g q | g | G “ n is a probability distribution on the | ¨ | G -sphere t| g | G “ n u . From this fact we derive the estimate ÿ | g | G “ n p´ p g log p g q “ ÿ | g | G “ n p´p µ g q n q log p µ g q n qq“ q n H ` p µ g q | g | G “ n ˘ ´ q n log q n ď q n log | B G p e G , n q| ´ q n log q n ď cnq n ´ q n log q n . whenever q n ‰ .Finally, some elementary calculus gives ´ t log t ď mt ` e ´ m ´ for any t, m ą . Let k ą be large and fixed, and for each n ě apply this bound with t : “ q n and m : “ kn . It gives ÿ n ě p´ q n log q n q ď k ÿ n ě nq n ` ÿ n ě e ´ kn ´ . Combining this with the previous estimate, we obtain ÿ g ‰ e G p´ p g log p g q “ ÿ n ě ÿ | g | G “ n p´ p g log p G q ď p c ` k q ÿ n ě nq n ` ÿ n ě e ´ kn ´ . By choosing k large enough we may make the last term here less than ε , so thiscompletes the proof of the first inequality.We obtain the second part of the lemma by applying that first inequality to thevalues p g : “ µ t α “ g u . Corollary 2.2.
Suppose that G and H are finitely generated groups, that p X, µ, T q is a G -system, and that α : G ˆ X ÝÑ H is an integrable cocycle over T . Forevery g P G and every ε ą there exists δ ą for which the following holds: forany measurable U Ď X ,if µ p U q ă δ then H µ p α g ; U q ă ε. Proof.
First, any sufficiently small δ satisfies µ p U q ă δ ùñ H µ p U q ă ε { .
13n the other hand, since α g is integrable, for any η ą there is a δ ą suchthat µ p U q ă δ ùñ ż U | α p g, x q| H µ p dx q ă η. By a special case of Lemma 2.1, we may choose C ă 8 so that this turns into H µ | U p α g q ď Cη { µ p U q ` , where C does not depend on the value of η .Combining these estimates gives µ p U q ă δ ùñ H µ p α g ; U q ă ε { ` Cη ` µ p U q ă ε { ` Cη ` δ. Choosing η ă ε { C and then ensuring that δ ă ε { , this completes the proof. Given a partial cocycle p α, U q over p X, µ, T q , the factor that it generates is thesmallest factor which contains U and with respect to which all of the partial ob-servables p α g , U X T g ´ U q are measurable. More explicitly, it is generated by the sets U g,h : “ t x P U X T g ´ U : α p g, x q “ h u for g P G and h P H , together with all their T -images. Notice that if we fix g and let h vary over H then the sets U g,h constitute a measurable partition of U X T g ´ U .Now suppose that Φ : p X, µ, T q p Y, ν, S q is an SOE, let U : “ dom Φ andlet V : “ img Φ . The compression of Φ is the constant comp p Φ q : “ ν p V q µ p U q . Let p α, U q be the partial cocycle associated to Φ , and p β, V q that associated to Φ ´ . Let U g,h be the sets defined above for the partial cocycle p α, U q , and let V h,g
14e their counterparts for p β, V q . For any g P G and h P H , the relation between α and β implies that x P U X T g ´ U and α p g, x q “ h ðñ x P U X T g ´ U and Φ p T g x q “ S h Φ p x qðñ Φ p x q P V X S h ´ V and β p h, Φ p x qq “ g. Therefore Φ p U g,h q “ V h,g for all g and h , and Φ p T g A q “ S h Φ p A q @ A Ď U g,h . (4)The next lemma is the first definite step in the direction of Theorem D. Lemma 3.1.
Let A be a factor of p X, µ, T q with respect to which p α, U q is mea-surable. Then there are • a factor map π : p X, µ, T q ÝÑ p X , µ , T q which generates A modulo µ , • another factor map ξ : p Y, ν, S q ÝÑ p Y , ν , S q , • and a SOE Φ : p X , µ , T q ÝÑ p Y , ν , S q such that the diagram p X, µ, T q / / Φ / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , T q / / Φ / / p Y , ν , S q commutes in the following sense: dom Φ “ π ´ p dom Φ q , and Φ ˝ p π | dom Φ q “ ξ ˝ Φ almost surely on this set.Proof. Let A : “ A X U , let C : “ Φ r A s , and let C be the factor of p Y, ν, S q generated by C .We now show that C X V “ C . The inclusion Ě is obvious. For the reverse, letus show that C is generated as a σ -algebra by a family of sets whose intersectionswith V are all members of C . In particular, let A P A , let C : “ Φ p A q , and let h P H : we will show that D : “ S h ´ C X V still lies in C . Since C Ď V , we have15 Ď S h ´ V X V . This right-hand set is partitioned into the subsets V h,g “ Φ p U g,h q , g P G , and these are all members of Φ r A s “ C by our assumption that p α, U q is A -measurable. Therefore D “ ď g P G D X V h,g “ ď g P G S h ´ C X V h,g “ ď g P G S h ´ ` C X S h V h,g ˘ “ ď g P G S h ´ ` Φ p A X T g U g,h q ˘ “ ď g P G Φ p T g ´ A X U g,h q , using (4) for the fourth and fifth equalities. This is explicitly a member of Φ r A s “ C , as required.Next, since p X, µ q and p Y, ν q are standard Borel, we may choose factor maps π : p X, µ, T q ÝÑ p X , µ , T q and ξ : p Y, ν, S q ÝÑ p Y , ν , S q which generate A modulo µ and C modulo ν , respectively. Since U P A Ď A and V P C Ď C ,there are measurable subsets U Ď X and V Ď Y such that U “ π ´ U and V “ ξ ´ V modulo negligible sets. Since A X U “ A and C X V “ C , itfollows that A “ π ´ r B U s modulo µ and C “ ξ ´ r B V s modulo ν , respectively.Therefore the set-mapping Φ r¨s : A ÝÑ C defines a measure-algebra equivalence from B U modulo µ to B V modulo ν .Since U and V are standard Borel, this arises from a measurable bijection Φ : U ÝÑ V . Now a simple diagram-chase shows that this fits into the desiredcommutative diagram. As remarked in the introduction, there is a close relationship between stable orbitequivalence of systems and measure equivalence of the acting groups: [Fur99,Theorem 3.3]. The main results of the present paper concern entropy, which is aproperty of the systems rather than the groups, so our point of view emphasizesthe former. However, some of the results we need are already known in the studyof measure equivalence, including most of those in this subsection.The first such result is a general procedure for extending a partial cocycle toa full cocycle. This fact can easily be extracted from the proof of the equivalencebetween stable orbit equivalence and measure equivalence, but for completenesswe include a proof purely in terms of cocycles.16 roposition 3.2. If p X, µ, T q is an ergodic G -system and p α, U q is a non-trivial H -valued partial cocycle over it, then there is a cocycle σ : G ˆ X ÝÑ H suchthat α “ σ | U .If τ : G ˆ X ÝÑ H is another cocycle satisfying α “ τ | U , then σ and τ arecohomologous over p X, µ, T q .Proof. Part 1. Let us enumerate G “ t g “ e G , g , g , . . . u . Since µ p U q ą and the system is ergodic, we have µ ´ ď g P G T g ´ U ¯ “ that is, U meets almost every T -orbit. Therefore for a.e. x P X there is a minimal n P N such that T g n p x q P U . This choice of g n defines a measurable function γ : X ÝÑ G such that T γ p x q p x q P U for a.e. x . We call it the U -return map .We now define σ p g, x q by • moving both x and T g x into the set U using the U -return map, and then • taking the value of α that connects those two new points.To be precise, this means that σ p g, x q : “ α ` γ p T g x q gγ p x q ´ , T γ p x q p x q ˘ . To see that this is well-defined, observe that the definition of γ gives T γ p x q p x q P U and also T γ p T g x q gγ p x q ´ ` T γ p x q p x q ˘ “ T γ p T g x q p T g x q P U, and so T γ p x q p x q P U X T p γ p T g x q gγ p x q ´ q ´ U “ dom ` α ` γ p T g x q gγ p x q ´ , ¨ ˘˘ . A simple check using the cocycle equation for α shows that the new map σ also satisfies the cocycle equation: σ p gk, x q “ α ` γ p T gk x q gkγ p x q ´ , T γ p x q p x q ˘ “ α ` γ p T gk x q gγ p T k x q ´ ¨ γ p T k x q kγ p x q ´ , T γ p x q p x q ˘ “ α ` γ p T gk x q gγ p T k x q ´ , T γ p T k x q p T k x q ˘ ¨ α ` γ p T k x q kγ p x q ´ , T γ p x q p x q ˘ “ σ p g, T k x q σ p k, x q . σ extends α , because if x P U X T g ´ U then γ p x q “ γ p T g x q “ e G (recalling that we put e G first in our enumeration of G ), and so σ p g, x q “ α ` γ p T g x q gγ p x q ´ , T γ p x q p x q ˘ “ α p g, x q . Part 2. If τ is another cocycle for which τ | U “ α , then the cocycle equationfor τ gives σ p g, x q “ α ` γ p T g x q gγ p x q ´ , T γ p x q p x q ˘ “ τ ` γ p T g x q gγ p x q ´ , T γ p x q p x q ˘ “ τ p γ p T g x q , T g x q ¨ τ p g, x q ¨ τ p γ p x q ´ , T γ p x q p x qq“ τ p γ p T g x q , T g x q ¨ τ p g, x q ¨ τ p γ p x q , x q ´ “ η p T g x q ´ ¨ τ p g, x q ¨ η p x q , where η p x q : “ τ p γ p x q , x q ´ is a measurable function from X to H . So σ ismanifestly cohomologous to τ . Remark . If the partial cocycle p α, U q satisfies an assumption of boundednessor integrability, then Proposition 3.2 gives no guarantee that its extension σ satis-fies the same assumption. We must therefore by quite careful in how we apply thisproposition to the study of SSOE or SOE . In the case of SSOE , an integrableextended cocycle is guaranteed by definition, but we sometimes need to performsome other manipulations on a cocycle first and then apply Proposition 3.2, socare is still necessary. ⊳ Now suppose that
Φ : p X, µ, T q p Y, ν, S q is a stable orbit equivalence froma free ergodic G -system to a free ergodic H -system. Let p α, U q and p β, V q be thepartial cocycles that describe Φ and Φ ´ . We now use Proposition 3.2 to constructa kind of ‘common extension’ of the two systems p X, µ, T q and p Y, ν, S q .This construction can be carried out starting from either p X, µ, T q or p Y, ν, S q .We begin by using the former. First, apply Proposition 3.2 to obtain a cocycle p α : G ˆ X ÝÑ H such that α “ p α | U . Now let p X : “ X ˆ H and let p µ be the σ -finite measure on this space which is the product of µ and counting measure.We define an infinite-measure-preserving action p T of G ˆ H on p p X, p µ q by setting p T p g,h q p x, k q : “ p T g x, p α p g, x q kh ´ q for g P G and h P H. p p X, p µ, p T q is ergodic. Indeed, if A Ď p X its invariant, thenthe action of H on the vertical fibres of p X forces A to have been lifted from X ,but an invariant set lifted from X must be negligible or co-negligible because T isergodic.Starting with p Y, ν, S q , the analogous construction uses an extension p β of β todefine a p G ˆ H q -action p S on p Y : “ Y ˆ G that preserves the product p ν of ν andcounting measure. Lemma 3.4.
The infinite-measure-preserving p G ˆ H q -systems p p X, p µ, p T q and p p Y , p ν, p S q are isomorphic, up to changing the measures by a constant multiple.Proof. Define p U : “ U ˆ t e H u Ď p X and p V : “ V ˆ t e G u Ď p Y , and let Φ : p U ÝÑ p V be the map that results from the obvious identification of p U with U and p V with V .The idea is that Φ should be a ‘part’ of the required isomorphism, and now p G ˆ H q -equivariance tells us how to extend it. Thus, for p x, k q P p X , choose p g, h q P G ˆ H so that p T p g,h q p x, k q P p U (we may do this almost surely by theergodicity of p T ), and let p Φ p x, k q : “ p S p g ´ ,h ´ q ` Φ p p T p g,h q p x, k qq ˘ . We must check that this is well-defined. Suppose that p g , h q P G ˆ H alsosatisfies p T p g ,h q p x, k q P p U , and let p g , h q : “ p g g ´ , h h ´ q . The assumption that p T p g,h q p x, k q P p U is equivalent to T g x P U and p α p g, x q kh ´ “ e G , and similarlyfor p g , h q . Combining these relations with the cocycle equation for p α , we obtain e G “ p α p g , x q ¨ k ¨ p h q ´ “ α p g , T g x q p α p g, x q kh ´ h ´ “ α p g , T g x q h ´ . Hence h “ α p g , T g x q , and so also g “ β p h , Φ p T g x qq , by (1). From this wededuce that Φ p p T p g ,h q p x, k qq “ ` Φ p T g x q , e G ˘ “ ` Φ p T g p T g x qq , e G ˘ “ ` S α p g ,T g x q p Φ p T g x qq , e G ˘ “ ` S h p Φ p T g x qq , e G ˘ “ ` S h p Φ p T g x qq , β p h , Φ p T g x qq e G g ´ ˘ “ p S p g ,h q p Φ p p T p g,h q p x, k qq . p S p g ,h q ´ ` Φ p p T p g ,h q p x, k qq ˘ “ p S p g ,h q ´ p S p g ,h q p Φ p p T p g,h q p x, k qq“ p S p g ´ ,h ´ q ` Φ p p T p g,h q p x, k qq ˘ , showing that the definition of p Φ p x, k q does not depend on which valid choice wemake of p g, h q .Analogous reasoning shows that p Φ is equivariant between the two p G ˆ H q -actions.Clearly p Φ | p U “ Φ , and for subsets of p U this amplifies the measure p µ by thefixed constant comp p Φ q . Since p Φ is equivariant and both of the systems p p X, p µ, p T q and p p Y , p ν, p S q are ergodic, this fact extends to the whole of p Φ . This shows that p Φ has the desired properties.Behind Lemma 3.4 lies a more conceptual fact: p p X, p µ, p T q and p p Y , p ν, p S q can beidentified with the measure coupling of G and H that arises from the given stableorbit equivalence, as in the proof of [Fur99, Theorem 3.3]. So far in this sectionwe have not assumed that Φ is a SOE or SSOE . However, if Φ is a SSOE , thenby definition we may choose the extended cocycles p α and p β to be integrable. Wetherefore obtain the following integrable analog of [Fur99, Theorem 3.3]. Thiscorollary is certainly already known to experts, but we record it explicitly for laterreference. Corollary 3.5.
If there exists a SSOE from a G -system to an H -system, then G and H are integrably measure equivalent. Given an SOE between ergodic actions of two groups, and also a finite-index sub-group of each group, one can construct a new SOE between ergodic actions ofthose subgroups. The construction is explained in this section in case the sub-groups are normal. Similar arguments can be carried out without the assumptionof normality, but extra technicalities arise which we do not address here. Many ofthe results we need can be found in [Fur99, Sections 2 and 3], up to the translationbetween SOE and measure equivalence: see, for instance, [Fur99, Example 2.9].If p X, µ, T q is a G system and G ď G is a subgroup, then T | G denotes therestriction of the action to G .Our first tool is the following simple lemma.20 emma 4.1. Let p X, µ, T q be an ergodic G -system, and let G E G have finiteindex. Then there is a finite measurable partition P of X into sets of equal mea-sure such that the ergodic components of the system p X, µ, T | G q are obtained byconditioning µ on the cells of P .Proof. Let g G , . . . , g k G be the distinct left cosets of G in G . Let A be the σ -algebra of T | G -invariant sets. It is a factor of the whole G -action, because G is normal in G . If A P A has positive measure, then G -invariance implies that ď g P G T g A “ k ď i “ T g i A. (5)This set is invariant for the whole G -action and has positive measure, so thatmeasure must equal by ergodicity. Therefore µ p A q ě { k .So all members of A either have measure zero or have measure at least { k ,and so A is atomic modulo negligible sets. Letting P be a set of atoms for A modulo negligible sets, we obtain that(i) the action T | G is ergodic inside each cell of P , and(ii) the action T permutes the cells of P , and must do so transitively becauseany union as in (5) has full measure in X .Conclusion (i) implies that the ergodic components of p X, µ, T | G q are obtainedby conditioning on the cells of P , and conclusion (ii) implies that all those cellshave the same measure. Lemma 4.2.
In the setting of the previous lemma, if G is amenable and P P P then h p µ | P , T | G q “ r G : G s ¨ h p µ, T q . Proof.
A standard calculation from the definition of KS entropy gives h p µ, T | G q “ r G : G s ¨ h p µ, T q . (6)On the other hand, since all cells of P have equal measure, the affinity of theentropy function gives h p µ, T | G q “ | P | ÿ P P P h p µ | P , T | G q . (7)21astly, all of the systems p µ | P , T | G q for P P P are conjugate-isomorphic. Indeed,if P, P P P , then we may choose g P G such that T g P “ P , and now thetransformation Ψ : “ T g sends the measure µ | P to the measure µ | P and satisfies Ψ ˝ T g “ T ϕ p g q ˝ Ψ for all g P G , where ϕ P Aut p G q is conjugation by g .Therefore all the summands on the right-hand side of (7) are equal, and soby (6) they must all be equal to r G : G s ¨ h p µ, T q .Now let p X, µ, T q and p Y, ν, S q be free ergodic G - and H -systems respec-tively. Let Φ : p X, µ, T q p Y, ν, S q be a SSOE or SOE , and let U : “ dom Φ and V : “ img Φ . Recall that we always assume G and H are infinite, and let G E G and H E H be normal subgroups of finite index.In this situation, we will construct an SSOE or SOE between a free ergodic G -system and a free ergodic H -system so that the new entropies and new com-pression are related to the old values in the following simple way. Proposition 4.3.
There are a free ergodic G -system p X , µ , T q and a free er-godic H -system p Y , ν , S q such that h p µ , T q “ r G : G s ¨ h p µ, T q and h p ν , S q “ r H : H s ¨ h p ν, S q , (8) and an SSOE (resp. SOE ) Φ : p X , µ , T q p Y , ν , S q such that comp p Φ q “ r H : H sr G : G s comp p Φ q . (9)This proposition enables one to deduce Theorem A for a pair of groups if itis known for a pair of finite-index subgroups. This is important in the case ofvirtually Euclidean groups, which are treated in the next section.The key to Proposition 4.3 is the infinite-measure-preserving p G ˆ H q -systemof Lemma 3.4. Let p α : G ˆ X ÝÑ H and p β : H ˆ Y ÝÑ G be extensions of α and β as given by Lemma 3.2, and let p p X, p µ, p T q and p p Y , p ν, p S q be the isomorphic p G ˆ H q -systems that appear in Lemma 3.4.22 roof. Step 1. We first construct a new G -system and a new H -system. In alater step we will restrict these to G and H and then obtain p X , µ , T q and p Y , ν , S q as ergodic components of those restrictions.Let G : “ G { G and H : “ H { H be the finite quotient groups. For elements g P G and h P H let g P G and h P H be their respective images.Now let X : “ X ˆ H and Y : “ Y ˆ G . Let r µ on X be the product of µ and the Haar measure on H , and define r ν on Y similarly. Let π X : X ÝÑ X and π Y : Y ÝÑ Y be the coordinate projections, so these are r H : H s -to- and r G : G s -to- , respectively.Define a G -action r T on p X , r µ q by r T g p x, h q : “ p T g x, p α p g, x q ¨ h q , and similarly define an H -action r S on p Y , r ν q by r S h p y, g q : “ p S h y, p β p h, y q ¨ g q . Then π X intertwines r T with T and π Y intertwines r S with S . Step 2.
Now let T : “ r T | G and S : “ r S | H . Doing so gives a free G -system p X , r µ, T q and a free H -system p Y , r ν, S q . These systems need not be ergodic.We set the issue of ergodicity aside for now, and next construct an SOE be-tween these systems. Let U : “ U ˆ t e H u and V : “ V ˆ t e G u , and observe that r µ p U q “ r H : H s µ p U q and r ν p V q “ r G : G s ν p V q . (10)Define r Φ : U ÝÑ V by r Φ p x, e H q : “ p Φ p x q , e G q . This is an SOE from p X , r µ, T q to p Y , r ν, S q . To see this, suppose that p x, e H q P U and g P G are such that T g p x, e H q “ p T g x, p α p g, x qq P U . Since x, T g x P U , we have p α p g, x q “ α p g, x q . Now the following both hold:(i) The points x and T g x lie in the same class of R T X p U ˆ U q , and hence their Φ -images lie in the same class of R S X p V ˆ V q , because Φ is an SOE.(ii) Since T g p x, e H q P U , we must have α p g, x q “ e H , and hence α p g, x q P H .Therefore the points Φ p x q and Φ p T g x q “ S α p g,x q Φ p x q actually lie in thesame H -orbit, not just the same H -orbit.23hese conclusions show that r Φ maps the classes of R T X p U ˆ U q into classes of R S X p V ˆ V q . By the symmetry of the construction, the same holds in reverse,and so r Φ is an SOE as required.Observe that the calculation (10) gives comp p r Φ q “ r ν p V q r µ p U q “ r H : H sr G : G s comp p Φ q , where we use the measures r µ and r ν on our two new systems. Step 3.
We will now replace r µ and r ν with ergodic measures so as to preservethe properties obtained above.This relies on the following observation. Let us identify G and G with thecorresponding subgroups in the first coordinate of G ˆ H , and similarly for H and H . Then the space p X , r µ q may be identified with a fundamental domain for theaction p T | H on the infinite measure space p p X, p µ q , and so p X , r µ, T q may be iden-tified with the factor of p p X, p µ, p T | G q consisting of p T H -invariant sets. Similarly, p Y , r ν, S q may be identified with the p S G -invariant factor of p p Y , p ν, p S | H q .As a result, there is a measure-preserving p G ˆ H q -action on p X , r µ q givenby the quotient of the full p G ˆ H q -system p p X, p µ, p T q , and T is the restriction ofthat p G ˆ H q -action to G . This p G ˆ H q -action on p X , r µ q is ergodic, becausethe infinite-measure-preserving system above it is ergodic. On the other hand, G – G ˆ t e H u is normal in G ˆ H , because G is normal in G . We maytherefore apply Lemma 4.1 to the inclusion of T into this larger ergodic p G ˆ H q -action. It tells us that the G -system p X , r µ, T q has some finite number, say n , ofergodic components, and each of them is obtained by conditioning on an invariantset of measure { n . Let P be the partition of X consisting of these components.An analogous argument gives a finite partition Q of p Y , r ν q into equal-measureergodic components for S ; let m be the number of these.Crucially, we can now show that n “ m . Since p X , r µ q is the quotient of p p X, p µ q by the action p T | H , we may identify P with the partition of p p X, p µ q intoergodic components for the combined action p T | G ˆ H . Similarly, Q may be iden-tified with the partition of p p Y , p ν q into ergodic components for the combined action p S | G ˆ H . But those two actions are isomorphic up to a constant change of mea-sure, by Lemma 3.4, and so they have the same numbers of ergodic components.To finish our construction, choose one of the ergodic components P P P forwhich r µ p P X U q ą . The restriction of P to U gives the ergodic decompositionof R T X p U ˆ U q up to negligible sets, and r Φ carries that restriction to the24rgodic decomposition of R S X p V ˆ V q . Therefore r Φ identifies P X U with Q X V for a unique cell Q P Q . Now let µ : “ r µ | P and ν : “ r ν | Q . Then therestriction Φ : “ r Φ | P X U defines a SOE from p X , µ , T q to p Y , ν , T q , andthese are a free ergodic G -system and a free ergodic H -system respectively.For these systems, we may calculate the entropy using Lemma 4.2. On theother hand, we observe that comp p Φ q just equals comp p r Φ q because n “ m . Notethat, although Φ is simply a restriction of r Φ to a subset, the equality n “ m isneeded for this second calculation because the measures have also been changed:from r µ and r ν to their restrictions µ and ν .The partial cocycles associated to Φ and Φ ´ are simply restrictions of thoseassociated to Φ and Φ ´ . Also, the new measures µ and ν have bounded Radon–Nikodym derivatives with respect to r µ and r ν respectively. Therefore Φ is anSSOE (resp. SOE ) if Φ has this property. This section proves Theorem B, which concerns a SOE or SSOE between ac-tions of Euclidean lattices. From this we deduce Theorem A in case G and H areboth virtually Euclidean: that is, they contain finite-index subgroups isomorphicto Euclidean lattices. By intersecting finitely many conjugates, one may assumethat those subgroups are normal. Let e , . . . , e d be the standard basis of Z d , andlet | ¨ | be the corresponding ℓ -norm on Z d .The special case of Euclidean lattices is important for two reasons. Firstly,we will make contact with the older notion of Kakutani equivalence for actionsof Euclidean lattices, which has been studied much more thoroughly than SSOE .I do not know whether these notions are actually equivalent. Secondly, our ap-proach to Theorem A in the remainder of the paper needs the assumption that G and H have super-linear growth, so it does not cover virtually cyclic groups. Wetherefore need the results of the present section to prove Theorem A in that case.For groups containing a finite-index copy of Z d with d ě , we end up with twoproofs of Theorem A, one in the present section and the other from the remainderof the paper.We start with Theorem B, which applies to Euclidean lattices themselves, andthen prove Theorem A for virtually Euclidean groups using Theorem B and Propo-sition 4.3.In the Euclidean case, Theorem B connects SOE and SSOE with the gen-eralization of Kakutani equivalence to Z d -actions developed in [Kat77, dJR84,25B92]. We use the definition of this property from [dJR84, Definition 3]: Definition 5.1.
Let M be a real p d ˆ d q -matrix. Two Z d -systems p X, µ, T q and p Y, ν, S q are M -Kakutani equivalent if there is an SOE Φ : p X, µ, T q p Y, ν, S q with the following properties:(i) dom Φ “ X , and(ii) if α : Z d ˆ X ÝÑ Z d is the cocycle describing Φ , then for any ε ą thereare N ε P N and A ε Ď X with µ p A ε q ą ´ ε such that, if v P Z d has | v | ě N , and x P A ε X T ´ v A ε , then | α p v, x q ´ M v | ď ε | v | . In their paper, del Junco and Rudolph refer to Φ as an ‘orbit injection’, ratherthan a ‘SOE’, and say that it ‘maps distinct orbits into distinct orbits’. They alsomake the explicit assumption that M is invertible with | det M | ě . However, theparagraph immediately following the proof of their Proposition 3 makes it clearthat this is what we call a SOE, and that the other parts of Definition 5.1 actuallyrequire that | det M | ě .It is helpful to know that part (ii) of Definition 5.1 can be replaced by thefollowing apparently weaker condition: (ii) For any ε ą there are N ε P N and A ε Ď X with µ p A ε q ą ´ ε such that,if ď i ď d , n ě N ε , and x P A ε X T ´ ne i A ε , then | α p ne i , x q ´ nM e i | ă εn. The condition that this holds for some basis in Z d is Condition 1 on p93 of [dJR84].The fact that it implies M -Kakutani equivalence is their Proposition 7.The first assertion of Theorem B reduces our work to the case of SSOE . Weisolate it as the following lemma. Lemma 5.2. If p X, µ, T q is a Z d -system, p Y, ν, S q is a Z D -system, and they areSOE , then the partial cocycles α and β which describe this SOE have extensionsto full cocycles Z d ˆ X ÝÑ Z D and Z D ˆ Y ÝÑ Z d which are still bounded. Inparticular, the systems are SSOE . roof. It suffices to show that any bounded Z D -valued partial cocycle over p X, µ, T q can be extended to a bounded cocycle Z d ˆ X ÝÑ Z D . Arguing coordinate-wiseit suffices to prove this when D “ . Thus, let p α, U q be a Z -valued partial cocy-cle over p X, µ, T q , and assume that | α p v, x q| ď C | v | for µ -a.e. x P U X T ´ v U ,for all v P Z d .For each x P X let D x : “ t v P Z d : T v x P U u , the U -return set of x . Since p X, µ, T q is ergodic and µ p U q ą , this D x isnonempty for almost every x . By removing a negligible set, we may assumethis holds for strictly every x .Now consider x P U , so P D x . Then the assumed boundedness of α isequivalent to the assertion that the map D x ÝÑ Z : v ÞÑ α x p v q is C -Lipschitz for the restriction of | ¨ | to D x . We may therefore apply a standardconstruction to extend it to a C -Lipschitz map from the whole of Z d to R , andthen apply some rounding to produce a Z -valued function. To be specific, for u P Z d , let us define σ x p u q : “ X min t α x p v q ` C } u ´ v } : v P D x u \ , where t ¨ u is the integer-part function. This is p C ` q -Lipschitz, where the extra‘ ’ allows for the rounding. It extends α x , and it satisfies the following slightlyextended cocycle identity: σ x p u ` w q “ σ x p u q ` σ T u x p w q “ α x p u q ` σ T u x p w q whenever x, T u x P U. (11)Finally, the cocycle equation tells us how to extend σ further to a function onthe whole of Z d ˆ X . For each x P X , choose some v P D x , and let σ x p u q : “ σ T v x p u ´ v q ´ σ T v x p´ v q . (12)A re-arrangement using equation (11) shows that this right-hand side does notdepend on v , so σ x p u q is well-defined. If x P U then we may use the choice v “ , which shows that σ does indeed extend σ . The new function σ x is still p C ` q -Lipschitz on Z d for each x because σ T v x has that property.27t remains to verify the cocycle identity for σ . Suppose that x P X and u, w P Z d , and choose v P D x . It follows that v ´ w P D T w x . Therefore, using these twopoints in the right-hand side of (12), we obtain σ x p u ` w q “ σ T v x p u ` w ´ v q ´ σ T v x p´ v q“ σ T v ´ w p T w x q p u ´ p v ´ w qq ´ σ T v ´ w p T w x q p´p v ´ w qq` σ T v x p w ´ v q ´ σ T v x p´ v q“ σ T w x p u q ` σ x p w q , as required. Proof of Theorem B.
By the preceding lemma, it suffices to assume that
Φ : p X, µ, T q p Y, ν, S q is an SSOE . By considering Φ ´ instead if necessary, we may assume that comp p Φ q ď .This SSOE between the systems implies an integrable measure equivalencebetween the two groups, by Corollary 3.5. As shown by Lewis Bowen in [Aus16,Theorem B.2], this requires that they have the same growth, and hence D “ d .Let U : “ dom Φ and V : “ img Φ , and let α : Z d ˆ Y ÝÑ Z d be an integrablecocycle such that p α, U q describes Φ .Since comp p Φ q ď , we have ν p V q ď µ p U q . Choose a measurable subset W Ď Y such that W Ě V and ν p V q{ ν p W q “ µ p U q . By [Fur99, Proposition2.7], Φ has an extension to an isomorphism r Φ between the relations R T and R S Xp W ˆ W q : that is, r Φ is a SOE which extends Φ , whose domain is the whole of X , and whose image is W . It has the same compression as Φ . Since dom r Φ “ X ,it is described by a cocycle σ : Z d ˆ X ÝÑ Z d such that σ | U “ α | U and suchthat σ x is an injection for a.e. x . It does not follow that σ is integrable, but since σ | U “ α | U , the second part of Proposition 3.2 promises that σ is cohomologous to α , say σ p v, x q “ α p v, x q ` γ p T v x q ´ γ p x q for some γ : X ÝÑ Z d .Next, since α is integrable, the cocycle equation and the pointwise ergodictheorem give that α p ne i , x q n “ n n ´ ÿ j “ α p e i , T je i x q ÝÑ v i : “ ż α p e i , x q µ p dx q as n ÝÑ 8 (13)28or µ -almost every x and for i “ , , . . . , d . Let M be the p d ˆ d q -matrix whosecolumns are the vectors v i .We now show that the SOE r Φ is an M -Kakutani equivalence for this M . Wehave guaranteed condition (i) by construction, and we finish the proof by showingcondition (ii) instead of (ii). Given ε ą , choose r ε ă 8 so large that the set B ε : “ t x : | γ p x q| ď r ε u has µ p B ε q ą ´ ε { . Now choose N ε so large that r ε ă εN ε { and so that the set C ε : “ x : | α p ne i , x q ´ nv i | ă εn { @ n ě N ε @ i “ , , . . . , d ( has µ p C ε q ą ´ ε { ; this is possible because of (13). Finally, let A ε : “ B ε X C ε .Then µ p A ε q ą ´ ε , and for any n ě N ε and x P A ε X T ´ ne i A ε we obtain | σ p ne i , x q ´ nM e i | ď | α p ne i , x q ´ nv i | ` | γ p x q| ` | γ p T ne i x q| ă εn { ` r ε ă εn. Corollary 5.3.
The conclusion of Theorem A holds if G and H are virtually Eu-clidean.Proof. If G and H are strictly Euclidean, then Theorem B reduces this to thecorresponding result for Kakutani equivalence. By the explanation which followsthe proof of Proposition 3 in [dJR84], the matrix M constructed in the proof ofTheorem B must satisfy comp p Φ q “ comp p r Φ q “ | det M | . Now the desired result follows from the equation h p ν, S q “ h p µ, T q| det M | , which is recalled on the last page of [dJR84] (beware that this equation also ap-pears at the bottom of p91 of their paper, but written incorrectly). Del Junco and29udolph attribute this equation to an unpublished work of Nadler, but the spe-cial case of id -Kakutani equivalence is included as [HB92, Corollary 3], and thegeneral case is proved in the same way.Now suppose that G E G and H E H are finite-index subgroups isomorphicto Euclidean lattices. Let p X , µ , T q , p Y , ν , S q and Φ be the systems and SOEgiven by Proposition 4.3. Then the special case of Euclidean groups gives that h p ν , S q “ comp p Φ q h p µ , T q , and now the equations (8) and (9) turn this into the desired conclusion. Remark . Beyond Kakutani equivalence for Z d -actions, Kammeyer and Rudolphhave developed a very abstract notion of ‘restricted orbit equivalences’ betweenactions of discrete amenable groups: see [KR97, KR02]. I do now know whetherOE or SSOE are examples of restricted orbit equivalences, but if so then theirmachinery would have several consequences in our setting, such as an analog ofOrnstein theory. ⊳ Question . Is it true that SOE implies SSOE between actions of other finitelygenerated amenable groups? Does Kakutani equivalence imply either? ⊳ This section returns to the setting of general amenable-group actions. It derivesTheorem A from Theorem D. The more difficult work of proving Theorems C andD occupies the rest of the paper after this.Theorem D leads to Theorem A via the following.
Proposition 6.1.
Let p X, µ, T q / / Φ / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , T q / / Φ / / p Y , ν , S q e a commutative diagram whose rows are SOEs, whose left column is a factormap of free G -systems, and whose right column is a factor map of free H -systems.Then the relative entropies over those factor maps satisfy µ p dom Φ q ´ h p µ, T | π q “ ν p img Φ q ´ h p ν, S | ξ q . This result may already be known, but I have not found a suitable referencein the literature. It may be a consequence of Danilenko’s quite abstract resultsin [Dan01, Section 2], but it seems worth including a more classical proof. Asimple approach, suggested to me by Lewis Bowen, is based on the followinglemma.
Lemma 6.2.
Let π : p X, µ, T q ÝÑ p X , µ , T q be a factor map of free G -systems.Then there is a commutative diagram p X, µ, R q id X / / π (cid:15) (cid:15) p X, µ, T q π (cid:15) (cid:15) p X , µ , R q id X / / p X , µ , T q in which R and R are single transformations and id X and id X are OEs (equiva-lently, R and T have the same orbits and R and T have the same orbits).Proof. By the main result of [CFW81], there is a single µ -preserving transforma-tion R on X which has the same orbits as the action T . Since p X , µ , T q is free,this implies the existence of a unique cocycle α : X ÝÑ G such that R x “ p T q α p x q x for x P X . The proof is completed by defining Rx : “ T α p π p x qq x for x P X. Lemma 6.2 enables us to convert G - and H -actions into Z -actions, for whichstable orbit equivalence is easier to understand. For Z -actions, stable orbit equiv-alence is simply an orbit equivalence between induced transformations, whoseentropy is computed by Abramov’s formula.31 roof of Proposition 6.1. Let U : “ dom Φ , U : “ dom Φ , V : “ img Φ and V : “ img Φ . Our assumptions include that U “ π ´ U and V “ ξ ´ V .First we invoke Lemma 6.2 on the left-hand side of the diagram in the state-ment of Proposition 6.1. This produces the larger diagram p X, µ, R q id X / / π (cid:15) (cid:15) p X, µ, T q / / Φ / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , T q id X / / p X , µ , T q / / Φ / / p Y , ν , S q . By [RW00, Theorem 2.6], the left-hand square above gives the equality h p µ, R | π q “ h p µ, T | π q . (14)Now composing the rows of this diagram, it collapses to p X, µ, R q / / Φ / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , R q / / Φ / / p Y , ν , S q . In view of (14), it suffices to show that this diagram implies the equality µ p U q ´ h p µ, R | π q “ ν p V q ´ h p ν, S | ξ q : that is, we have reduced the desired proposition to the case G “ Z .Applying Lemma 6.2 in the same way on the right-hand side of the diagram,we may reduce to the case in which G “ H “ Z , and so T and S may be regardedas single transformations. However, in this case Φ (resp. Φ ) is an OE betweenthe induced transformations T U and S V (resp. T U and S V ), and so another appealto [RW00, Theorem 2.6] gives h ` µ | U , T U ˇˇ π | U ˘ “ h ` ν | V , S V ˇˇ ξ | V ˘ . Finally, Abramov’s formula for the entropy of induced transformations [Abr59]and the Abramov-Rokhlin formula for the entropy of an extension [AR62] give h ` µ | U , T U ˇˇ π | U ˘ “ h ` µ | U , T U ˘ ´ h ` µ U , T U ˘ “ µ p U q ´ ` h p µ, T q ´ h p µ , T q ˘ “ µ p U q ´ h p µ, T | π q , and similarly for h ` ν | V , S V ˇˇ ξ | V ˘ . 32 ompleted proof of Theorem A, given Theorem D. First suppose that either G or H has linear growth. Lewis Bowen has shown in [Aus16, Theorem B.2] thatgrowth type is an invariant of integrable measure equivalence, so this implies thatthey both have linear growth, and hence they are both virtually Z . So in this casethe result follows from Section 5.Now suppose that both groups have super-linear growth. Let Φ : p X, µ, T q p Y, ν, S q be either a SSOE or a SOE , and let c : “ comp p Φ q . In this caseTheorem D gives a positive-measure subset U Ď dom Φ and a diagram of theform p X, µ, T q / / Φ | U / / π (cid:15) (cid:15) p Y, ν, S q ξ (cid:15) (cid:15) p X , µ , T q / / Φ / / p Y , ν , S q , where h p µ , T q ă ε and all the systems are free. It follows that c “ ν p Φ p U qq µ p U q “ ν p img Φ q µ p dom Φ q . We now combine Proposition 6.1 with Ward and Zhang’s generalization of theAbramov–Rokhlin formula to extensions of amenable-group actions [WZ92, The-orem 4.4]. This gives h p ν, S q ě h p ν, S | ξ q“ c h p µ, T | π q“ c ` h p µ, T q ´ h p µ , T q ˘ ě c ` h p µ, T q ´ ε ˘ . Since ε ą was arbitrary, it follows that h p ν, S q ě c h p µ, T q , and the reverse inequality holds by symmetry. Most of the rest of the paper will go towards proving Theorem D. The next step isto introduce some more kinds of structure that will be used during the proof.33 .1 Graphings
Let p X, µ, T q be a G -system and R T Ď X ˆ X its orbit equivalence relation. Inthis setting, we need some definitions related to graphings and their costs. Graph-ings go back to Adams’ paper [Ada90], and cost to Levitt’s work [Lev95]. Theseconstructions have since become very important to the study of Borel equivalencerelations: see, for instance, Gaboriau’s survey [Gab02].In order to study integrable orbit equivalence, we need to work with graph-ings that are always defined with reference to the given G -action, and then with amodified notion of cost that accounts for the lengths of elements of G . We there-fore adjust the older definitions in the following way. A T -graphing is a family Γ “ p A g q g P G of measurable subsets of X indexed by G satisfying A g ´ “ T g A g @ g P G. (15)The associated graphing in Levitt’s sense is the family of partial maps T g | A g : A g ÝÑ T g A g .The vertex set of a graphing Γ is Vert p Γ q : “ Ť g A g , and Γ is nontrivial ifthis set has positive µ -measure. If Vert p Γ q “ V , we may regard Γ as placing thestructure of a graph on each of the equivalence classes in R T X p V ˆ V q , where x and T g x are joined by an edge if x P A g . Condition (15) is equivalent to this setof edges being symmetric, so we may regard this graph as undirected.The equivalence relation generated by a T -graphing Γ is the smallest Borelequivalence relation which contains p x, T g x q whenever g P G and x P A g . It isdenoted by R Γ . Definition 7.1. A T -graphing Γ is orbit-wise connected if R Γ X p V ˆ V q “ R T X p V ˆ V q for some V Ď Vert p Γ q with µ p Vert p Γ qz V q “ . Equivalently, this asserts thatfor µ -a.e. x P Vert p Γ q , the edges of R Γ define a connected graph on the set T G x X V . The factor of p X, µ, T q generated by the graphing Γ “ p A g q g is simply thesmallest factor which contains all the sets A g . We write h p µ, T, Γ q for the KSentropy of this factor. 34 .2 Graphings and partial cocycles Now suppose that p α, U q is an H -valued partial cocycle over p X, µ, T q and that Γ “ p A g q g is a T -graphing. Let V : “ Vert p Γ q , and assume that V Ď U . In viewof the relation (15), this impliesboth A g Ď U and A g ´ “ T g A g Ď U @ g P G, so in fact A g Ď U X T g ´ U for every g . We may therefore define the restrictionof α to Γ to be the restriction of α to the subset tp g, x q : g P G and x P A g u . Denote it by α | Γ . If Γ is the ‘na¨ıve’ graphing defined by A g : “ V X T g ´ V forevery g , then this agrees with our previous definition of α | V . The factor generated by α | Γ is the factor A generated by all the partial observables p α g | A g , A g q , g P G ,and its entropy is h p µ, T, α | Γ q : “ h p µ, T, A q . Lemma 7.2. If Γ is an orbit-wise connected T -graphing, and V : “ Vert p Γ q ,then the factor generated by α | Γ contains the factor generated by p α | V , V q up tonegligible sets.Proof. Let A be the factor generated by α | Γ . It contains every A g , so it containstheir union V , and so it contains all of the intersections V X T g ´ V (although theseneed not be equal to A g for any g ).Now fix g P G and h P H , and consider the subsets V g,h : “ t x P V X T g ´ V : α p g, x q “ h u . As g and h vary, these generate the σ -algebra of p α | V , V q . Since Γ is orbit-wiseconnected, we may remove a negligible set so that a point x P V X T g ´ V lies in V g,h if and only if there is a factorization g “ g k g k ´ ¨ ¨ ¨ g such that T g i ´ ¨¨¨ g x P A g i @ i “ , . . . , k and α p g, x q “ α p g k , T g k ´ ¨¨¨ g x q ¨ ¨ ¨ α p g , x q “ h. g , . . . , g k P G , and similarly for the sequence of elements α p g , x q , . . . , α p g k , T g k ´ ¨¨¨ g x q .Therefore we have expressed V g,h is a countable union of further subsets all ofwhich manifestly lie in the factor generated by α | Γ . Lemma 7.3.
Suppose that p α, U q is a partial cocycle over p X, µ, T q and that Γ “ p A g q g is a T -graphing for which Vert p Γ q Ď U . Then h p µ, T, α | Γ q ď ÿ g P G H µ p A g q ` ÿ g P G µ p A g q H µ | Ag p α g q . Proof.
This is a simple application of equation (3): h p µ, T, α | Γ q ď ÿ g P G h p µ, T, p α g | A g , A g qq ď ÿ g P G H µ p α g ; A g q“ ÿ g P G H µ p A g q ` ÿ g P G µ p A g q H µ | Ag p α g q . In combination, the previous two lemmas allow one to control the entropyof the factor generated by p α | V , V q using any choice of orbit-wise connectedgraphing with vertex set V . A careful choice of that graphing can give a bet-ter upper bound than a more na¨ıve estimate in terms of the partial observables p α g , V X T g ´ V q .The next definition gives our modified notion of cost. Definition 7.4.
The | ¨ | G -cost of a graphing Γ “ p A g q g is C |¨| G p Γ q : “ ÿ g P G | g | G ¨ µ p A g q . This differs from Levitt’s definition by the presence of | g | G as a weightingfactor.The | ¨ | G -cost will be the basis of several estimates later in the paper. Simplestamong these is the following. Lemma 7.5.
For every ε ą there is a C ε ă 8 such that for any T -graphing Γ we have h p µ, T, Γ q ď H µ p A e G q ` C ε ¨ C |¨| G p Γ q ` ε. roof. This follows from the bound h p µ, T, Γ q ď ÿ g P G H µ p A g q “ ÿ g P G r´ µ p A g q log µ p A g q ´ µ p X z A g q log µ p X z A g qs and Lemma 2.1.Our principal result about graphings and | ¨ | G -cost is the following, whichgives us great flexibility in finding low-cost T -graphings that are still ‘large’ inthe sense of orbit-wise connectedness. Proposition 7.6 (Existence of low-cost graphings) . Let G be a finitely-generatedamenable group of super-linear growth and p X, µ, T q a free ergodic G -system.Let U Ď X have positive measure, and let ε ą . Then there is a nontrivialorbit-wise connected T -graphing Γ such that Vert p Γ q Ď U, µ p Vert p Γ qq ă ε and C |¨| G p Γ q ă ε. This proposition will be proved in the next section.
This section culminates in the proof of Proposition 7.6. First we give two subsec-tions to some preparatory results. Let p X, µ, T q be a free ergodic G -system. The following nomenclature is not standard, but will be useful in the sequel.
Definition 8.1.
Let p X, d q be a metric space and r ą . An r -skeleton of X isa connected graph p V, E q in which V is an r -dense subset of X (that is, everyelement of X lies within distance r of some element of V ). Its d -weight is thequantity wt d p V, E q “ ÿ uv P E d p u, v q P r , `8s . Lemma 8.2.
Let p X, d q be a compact metric space, let r ą , and let p V, E q bean r -skeleton of p X, d q with d -weight w ă 8 . Then any subset Y Ď X has a p r q -skeleton of d -weight at most w ` r | V | . roof. Let W : “ t v P V : d p v, Y q ă r u . Since V is r -dense in the wholeof X , one must have B r p W q Ě Y , where B r p W q is the union of all open r -balls centred at points of W . For each w P W , pick y w P Y X B r p w q , and let V Y : “ t v w : w P W u . Since B r p V Y q Ě B r p W q , the set V Y is p r q -dense in Y .By removing edges from E if necessary, we may assume that it is a span-ning tree of V . Then, since V has a spanning tree with d -weight w , its furthersubset W Ď V has a spanning tree with d -weight at most w : this is the clas-sical lower bound of { for the Steiner ratio of a general metric space (see, forinstance, [Cie01, Chapter 3]). Let E Ď ` W ˘ be a spanning tree of W with wt d p W, E q “ ÿ ww P E d p w, w q ď ÿ xy P E d p x, y q “ w. Let E Y : “ t v w v w : ww P E u . Now p V Y , E Y q is a p r q -skeleton of Y , and wt d p V Y , E Y q ď wt d p W, E q ` r | E | ď w ` r p| W | ´ q ď w ` r | V | , using the fact that, in a tree such as p W, E q , one has | E | “ | W | ´ .Now let G be a finitely generated amenable group and d G a right-invariantword metric on it, as before. Given ε, r ą , let us say that a subset F Ď G is p ε, r q -Følner if | F | ă 8 and |p B G p r q ¨ F qz F | ď ε | F | , where we abbreviate B G p e G , r q “ : B G p r q . The amenability of G asserts that p ε, r q -Følner sets exist for every ε and r .The use of two parameters, ε and r , in specifying the Følner condition is some-what redundant, but in some of the proofs that follow it is convenient to be able tomanipulate them separately.We also need our Følner sets to satisfy another condition. Given E Ď G and r ą , we say E is r -connected if for any g, h P E there is a finite sequence g “ g , g , . . . , g m “ h with g i P E and d G p g i , g i ` q ď r for every i “ , , . . . , m ´ . Such a sequenceis called an r -path , and the integer m is its length . A set is connected if it is -connected. Lemma 8.3. If G is amenable, then for every ε, r ą it has an p ε, r q -Følner setwhich is connected. roof. Step 1. Let η : “ ε {| B G p r q| . Let F be an p η, r q -Følner set, and let F “ F Y ¨ ¨ ¨ Y F k be the partition of F into maximal p r q -connected subsets. Then we must have i ‰ j ùñ B G p r q F i X B G p r q F j “ H , and therefore | B G p r q F z F || F | “ k ÿ i “ | B G p r q F i z F i || F i | ¨ | F i || F | . Since the left-hand side of this equation is at most η , and the right-hand side is anaverage weighted by the factors | F i |{| F | , there must be some i ď k for which | B G p r q F i z F i || F i | ď η. (16)So F i is a p r q -connected p η, r q -Følner set. Step 2.
Now let E : “ B G p r q F i . If g, h P F i and d G p g, h q ď r , then there isa -path of length at most r from g to h in G , by the definition of the word metric d G . The first r elements of that path must be contained in B G p g, r q , and the last r elements must be contained in B G p h, r q , so the whole path is contained in E .Since F i is p r q -connected, it follows that E is connected.On the other hand, we have B G p r q ¨ E “ p B G p r q ¨ F i q Y p B G p r q ¨ p E z F i qq , and the first set in this right-hand union is just E again. Therefore |p B G p r q ¨ E qz E | ď | B G p r q ¨ p E z F i q| ď | B G p r q|| E z F i | . By (16), this is at most η | B G p r q|| F i | ď ε | E | , so E is p ε, r q -Følner.The main results of this section apply to groups of super-linear growth. Curi-ously, their proofs seem to require the following fact from geometric group theory. Proposition 8.4. If G is a finitely-generated group of super-linear growth, thenits growth is at least quadratic: there is a constant c ą such that | B G p r q| ě c r @ r ě . Lemma 8.5 (Skeleta for Følner sets) . If G is an amenable group of super-lineargrowth, then there is a constant c with the following property. For any r ě , if F is a connected p , r q -Følner set, then it has a p r q -skeleton p V, E q satisfying wt d G p V, E q ď c | F |{ r. Proof.
Let c be the constant given by Proposition 8.4. Let V Ď F be a maxi-mal p r q -separated subset, chosen so that it contains e G . The standard volume-comparison argument gives | V || B G p r q| “ ˇˇˇ ď g P V B G p g, r q ˇˇˇ ď | B G p r q ¨ F | ď | F | ùñ | V | ď | F || B G p r q| . Now consider the graph on V in which two points form an edge if the distancebetween them is at most r . This graph is connected, by the connectedness of F and the maximality of V . It therefore contains a spanning tree, whose edge-set isa family E of | V | ´ pairs of points in V . This gives the bound wt d G p V, E q ď r | E | ă r | V | ď r | F || B G p r q| ď r | F | c r “ c | F |{ r. The above lemma and Lemma 8.2 immediately combine to give the following.
Corollary 8.6 (Skeleta for subsets of Følner sets) . If G is an amenable groupof super-linear growth, then there is a constant c with the following property. If r ą , F is a connected p , r q -Følner set, and A Ď F , then A has a p r q -skeleton p V, E q satisfying wt d G p V, E q ď c | F |{ r. .2 Rokhlin subrelations We now return to the G -system p X, µ, T q . If x P X and A is a finite subset of theorbit T G p x q , then we say A is p ε, r q -Følner or r -connected if this holds for itspre-image in the group: that is, for the set t g P G : T g x P A u . Since the action is free, this pre-image has the same finite cardinality as A . If wereplace x with a different point T h x in the same orbit, then this pre-image of A changes by right-translation by h ´ . This does not affect the properties of being p ε, r q -Følner or r -connected, so those properties really depend only on the orbit T G p x q and the set A , not on the particular reference point x . Definition 8.7.
Let ε, r ą . A subrelation R Ď R T is p ε, r q -Rokhlin if it is aBorel equivalence relation, all its equivalence classes are finite and connected,and µ t x : r x s R is p ε, r q -Følner u ą ´ ε. This definition has many predecessors in the literature, but usually withoutrequiring connectedness. That additional demand adapts it to our present needs.
Lemma 8.8. If p X, µ, T q is ergodic and atomless then R T has an p ε, r q -Rokhlinsubrelation for every ε, r ą .Proof. According to one of the key results of [CFW81], R T may be written as Ť n ě R n for some increasing sequence R Ď R Ď ¨ ¨ ¨ Ď of Borel equivalencerelations with finite classes.For each i , define R i Ď R i by R i “ p x, y q P R i : x and y lie in the same connected component of r x s R i ( . These R i ’s are Borel equivalence relations for which every class r x s R i is finiteand connected. Also, their union is still equal to R T . To see this, let x P X and g P G . There is a finite -path e “ g , g , . . . , g k “ g in G . Since R T “ Ť n ě R n , for each i “ , , . . . , k ´ we have p T g i x, T g i ` x q P R n for all sufficiently large n, p T g i x, T g i ` x q P R n for all sufficiently large n, since d G p g i , g i ` q “ for each i . Hence, by transitivity, p x, T g x q P R n for allsufficiently large n .Finally, ż | T B G p r q pr x s R n qzr x s R n ||r x s R n | µ p dx q ď ÿ g P B G p r q ż | T g pr x s R n qzr x s R n ||r x s R n | µ p dx q“ ÿ g P B G p r q ż |t y P r x s R n : T g y R r x s R n u||r x s R n | µ p dx q“ ÿ g P B G p r q µ t x : p x, T g x q R R n u . This tends to as n ÝÑ 8 because R T “ Ť n ě R n . By Chebychev’s inequality,this implies that µ t x : r x s R n is p ε, r q -Følner u ą ´ ε for all sufficiently large n . We are ready to prove Proposition 7.6. The required T -graphing will be built asa union of a sequence of T -graphings given by the following lemma. Given twomeasurable subsets U, V Ď X , we say that V is p T, r q - dense in U if T B G p r q V Ě U. Lemma 8.9. If G is an amenable group of super-linear growth, then there is aconstant c ă 8 with the following property. Let p X, µ, T q be a free G -system andlet U Ď X have positive measure. Let ă ε ă µ p U q and r ă 8 . If R Ď R T is an p ε, r q -Rokhlin subrelation, then there is a T -graphing Γ “ p A g q g with thefollowing properties:i) V : “ Vert p Γ q Ď U ,ii) C |¨| G p Γ q ď c { r ,iii) R X p V ˆ V q “ R Γ , v) V is p T, r q -dense in the set U X t x : r x s R is p , r q -Følner u (in particular, this implies that µ p V q ą ).Proof. Let c be the constant from Corollary 8.6. Suppose that R is an p ε, r q -Rokhlin subrelation, and let X : “ t x : r x s R is p , r q -Følner u , so X is a union of R -classes and µ p X q ą ´ ε .Since all classes in R are finite, it has a transversal Y Ď X : that is, Y ismeasurable and contains a unique element from each class of R (see, for in-stance, [KM04, Example 6.1]). For each x P X let us write x for the uniqueelement of r x s R X Y .For y P Y X X , let B y : “ t g : T g y P U X r y s R u Ď G. Since y P X and B y is contained in t g : T g y P r y s R u , Corollary 8.6 gives a p r q -skeleton p W y , E y q for B y satisfying wt d G p W y , E y q ď c |r y s R |{ r. Clearly W y and E y may be chosen measurably in y .Now transport these skeleta from G back to X by setting W y : “ T W y p y q and E y : “ t T h y, T g y u : t h, g u P E y ( for y P Y X X . The result is a graph p W y , E y q on a subset of each class r y s R Ď X .Finally, define the T -graphing Γ “ p A g q g by setting A e G : “ t x P X : x P W x u and A g : “ t x P X : x P W x and t x, T g x u P E x u for g P G zt e G u . This is symmetric: if x P A g and we set x : “ T g x , then t x, T g x u “ t x , T g ´ x u P E x and so also x P W x and x P A g ´ .It remains to verify the four required properties.43) For each x P X we have W x “ T W x p x q Ď T B x p x q Ď U, by the definition of B x . Hence A g Ď U for each g .ii) To estimate the cost, first observe that we may write A g “ ď h P G T h y : y P Y X X and t h, gh u P E y ( . This is a disjoint union: if T h y “ T h y among the points allowed above,then this point lies in W y Ď r y s R by the definition of E y , and this impliesthat h “ h and y “ y because T is free and Y contains a unique elementin each class of R . Therefore C |¨| G p Γ q “ ÿ g P G | g | G ¨ µ p A g q“ ÿ g P G ÿ h P G | g | G ¨ µ t y P Y X X : t h, gh u P E y u“ ż Y X X ÿ t h,gh uP E y d G p h, gh q µ p dy q“ ż Y X X wt d G p W y , E y q µ p dy qď cr ż Y X X |r y s R | µ p dy q ď cr ż Y |r y s R | µ p dy q “ cr . iii) Observe that V “ ď g A g “ ď y P Y X X W y . Therefore R X p V ˆ V q “ ď y P Y X X W y ˆ W y , and this equals R Γ because all the graphs p W y , E y q are connected.iv) Lastly, if x P U X X , then B x is nonempty, and then W x is p r q -dense in B x by construction. This implies that V is p T, r q -dense in U X X .44 roof of Proposition 7.6. Let c be the constant from Lemma 8.9, and choose m P N so that ´ m ă ε { c . Also, shrink U if necessary so that ă µ p U q ă ε . Step 1.
For each n ě m , let R n Ď R T be a p ´ n , n q -Rokhlin subrelationwhich also has the property that the set C n : “ x : r x s R n is p , n q -Følner and T B G p n ` q x Ď r x s R n ( satisfies µ p C n q ą ´ ´ n ´ µ p U q .Now let W : “ U X č n ě m C n , so µ p W q ą µ p U q{ . Step 2.
Applying Lemma 8.9, let Γ m “ p A m,g q g be a T -graphing such that V : “ Vert p Γ m q Ď W has µ p V q ą , C |¨| G p Γ m q ď c { m , and R m X p V ˆ V q “ R Γ m . Step 3.
For each n ě m ` now apply Lemma 8.9 again to obtain a T -graphing Γ n “ p A n,g q g such that V n Ď V , C |¨| G p Γ n q ă c { n , R n X p V n ˆ V n q “ R Γ n , and V n is p T, n ` q -dense in V . The last conclusion can be obtained from part (iv)of Lemma 8.9 because V is contained in W and W is already contained in C n byconstruction. Observe that the choices of Γ n for n ą m depend on the choice of Γ m in Step 2, but not on each other. Step 4.
After finishing this recursion, define Γ “ p A g q g by A g : “ ď n ě m A n,g for each g P G. We will show that this has the desired properties. The symmetry property (15)holds for Γ because it holds for each Γ n . The vertex set of Γ is equal to V Ď U Vert p Γ m q “ V and then Vert p Γ n q Ď V for every n ě m ` . Thisimplies that µ p Vert p Γ qq ă ε , because µ p U q ă ε . Also, C |¨| G p Γ q ď ÿ n ě m C |¨| G p Γ n q ă ÿ n ě m c ´ n “ c ´ m ` ă ε. It remains to show that Γ is orbit-wise connected: that is, that R Γ “ R T X p V ˆ V q . Consider a pair of distinct points x, T g x P V . Choose the least n ě m whichsatisfies n ` ą | g | G . We will prove that p x, T g x q P R Γ by induction on this n .If n “ m , then | g | G ă m ` , and so certainly T B G p m ` q x X T B G p m ` q p T g x q ‰ H . Since also x, T g x P C m , the definition of C m now requires that r x s R m “ r T g x s R m .Since R m X p V ˆ V q “ R Γ m , it follows that p x, T g x q P R Γ m Ď R Γ .Now suppose that n ě m ` . Since V n is p T, n ` q -dense in V , there are h, k P G such that T h x, T k x P V n and d G p e G , h q , d G p g, k q ă n ` . By the inductive hypothesis, this implies that p x, T h x q , p T g x, T k x q P R Γ . On theother hand, the triangle inequality now gives d G p h, k q ă n ` , and so T B G p n ` q p T h x q X T B G p n ` q p T k x q ‰ H . Arguing as above, this implies that p T h x, T k x q P R n , and since these points arealso in V n this implies that p T h x, T k x q P R n X p V n ˆ V n q “ R Γ n Ď R Γ . Remark . It is easy to see that the above conclusion fails if G “ Z , so theassumption of super-linear growth is important. ⊳ Proof of the derandomization results
Proposition 9.1.
Let p σ, U q be a partial cocycle for which at least one of thefollowing holds:i) σ extends to an integrable cocycle G ˆ X ÝÑ H ;ii) p σ, U q is bounded.Then for every ε ą there is a δ ą such that the following holds. If Γ “ p A g q g P G is an orbit-wise connected T -graphing with Vert p Γ q Ď U , and both µ p A e G q ă δ and C |¨| G p Γ q ă δ , then h p µ, T, σ | Γ q ă ε .Proof. Case (i). Denote the extended cocycle G ˆ X ÝÑ H also by σ . From Γ we define a nearest-neighbour T -graphing Θ “ p B s q s P B G p q as follows. For each g P G , choose a word of length | g | G in the alphabet B G p q which evaluates to g ,say g “ s g,n s g,n ´ ¨ ¨ ¨ s g, where n : “ | g | G . Now define B g,i : “ T s g,i ´ ¨¨¨ s g, A g for i “ , , . . . , n, and finally let B s : “ ď g P G ď ď i ď| g | G s . t . s g,i “ s B g,i for each s P B G p q . This construction has the following two important features.(a) The new cost is bounded by the old cost: C |¨| G p Θ q “ ÿ s P B G p q µ p B s q ď ÿ g P G | g | G ÿ i “ µ p B g,i q“ ÿ g P G | g | G ÿ i “ µ p A g q “ ÿ g P G | g | G ¨ µ p A g q “ C |¨| G p Γ q ă δ. (b) Given the collection of sets A g for g P G and also the collection of partialfunctions σ p s, ¨q| B s for s P B G p q , they determine all the partial functions σ p g, ¨q| A g using the cocycle identity: σ p g, x q “ σ p s g,n , T s g,n ´ ¨¨¨ s g, x q ¨ ¨ ¨ σ p s g, , x q .
47n this formula, if x P A g , then x P B g, Ď B s g, , T s g, x P B g, Ď B s g, , . . . , T s g,n ´ ¨¨¨ s g, x P B s g,n ´ . Therefore the factor generated by σ | Γ is contained in the factor generatedby Γ and σ | Θ together.By property (b) above, we have h p µ, T, σ | Γ q ď h p µ, T, Γ q ` h p µ, T, σ | Θ q If δ is sufficiently small, then the first of these terms is at most ε { by Lemma 7.5.On the other hand, Lemma 7.3 gives h p µ, T, σ | Θ q ď ÿ s P B G p q H µ p σ p s, ¨ q ; B s q . By property (a) above, if δ is small enough, then we may apply Corollary 2.2 toeach summand on the right. This completes the proof in case (i). Case (ii).
In this case we can use Lemma 7.3 more directly: h p µ, T, σ | Γ q ď ÿ g P G H µ p A g q ` ÿ g P G µ p A g q H µ | Ag p σ p g, ¨ qq . (17)Since we are in case (ii), there is a finite constant C such that for each g therandom variable σ p g, ¨ q takes values in B H p C | g | G q almost surely. Since H isfinitely generated, we have log | B H p r q| “ O p r q for all r , and hence H µ | Ag p σ p g, ¨ qq ď log | B H p C | g | G q| “ O p C | g | G q “ O p| g | G q . Therefore the right-hand side of (17) is bounded by a constant multiple of ÿ g P G H µ p A g q ` ÿ g P G | g | G ¨ µ p A g q . By Lemma 7.5, the first term here may also be made arbitrarily small if µ p A e G q and C |¨| G p Γ q are sufficiently small. This completes the proof. Proof of Theorem C.
Given ε ą , apply case (i) of Proposition 9.1 to the cocycle σ with U : “ X . We obtain a δ ą for which the conclusion of that propositionholds. Now apply Proposition 7.6 to obtain a nontrivial orbit-wise connected T -graphing Γ such that Vert p Γ q Ď U , µ p Vert p Γ qq ă δ and C |¨| G p Γ q ă δ . The second48f these conditions implies that also µ p A e G q ă δ . Therefore, letting A be thefactor generated by σ | Γ , the choice of δ implies that h p µ, T, A q ă ε .Let V : “ Vert p Γ q . Since Γ is orbit-wise connected, Lemma 7.2 tells us that thepartial cocycle p σ | V , V q is also A -measurable. Now apply the first part of Proposi-tion 3.2 to the partial cocycle p σ | V , V q and the factor system of p X, µ, T q generatedby A , which must still be ergodic. That proposition gives an A -measurable cocy-cle τ : G ˆ X ÝÑ H such that σ | V “ τ | V . Since τ is A -measurable, its entropy isalso less than ε , and by the second part of Proposition 3.2 it is cohomologous to σ . Proof of Theorem D.
Fix ε ą . Cases (i) and (ii) of Theorem D correspondto cases (i) and (ii) of Proposition 9.1. Therefore in either case there is some δ ą for which the implication of that proposition holds. Having chosen this δ ,Proposition 7.6 gives a non-trivial orbit-wise connected T -graphing Γ “ p A g q g P G such that U : “ Vert p Γ q Ď dom Φ , µ p Vert p Γ qq ă δ and C |¨| G p Γ q ă δ. By the choice of δ this implies that h p µ, T, α | Γ q ă ε .Letting A be the factor generated by p α | U , U q , it now follows by Lemma 7.2that h p µ, T, A q ă ε . By enlarging A slightly if necessary, we may assume inaddition that it is generated by a factor map to another free G -system. FinallyLemma 3.1 produces the remaining objects with the properties asserted in Theo-rem D.
10 Further questions
Integrable measure equivalence was originally introduced in [BFS13] for actionsof lattices in isometry groups of hyperbolic spaces. It would be interesting toknow whether any classification of probability-preserving actions of such groupsfollows from the accompanying assumption of SSOE . Since these groups are notamenable, the Kolmogorov–Sinai entropy is not available as an invariant. How-ever, recent years have seen important progress in our understanding of entropy-like invariants for non-amenable groups. Question . If G and H are countable groups, does an SOE or SSOE be-tween a G -action and an H -action imply a relation between their Rokhlin en-tropies [Sewa, Sewb]? 49 uestion . If G and H are sofic groups, can one choose sofic approximationsto them in such a way that an SOE or SSOE between a G -action and an H -action imply a relation between their sofic entropies [Bow10a, KL11]? If G “ H is a free group, can one obtain a relation between f-invariants [Bow10b]? References [Abr59] L. M. Abramov. The entropy of a derived automorphism.
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URL: cims.nyu.edu/˜timcims.nyu.edu/˜tim