The geometry of model spaces for probability-preserving actions of sofic groups

Bowen's notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the `model spaces'. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

be the set of models which are 'good' according to this neighbourhood O.
Also, for each n, let d pVnq be the Hamming average of copies of d on X Vn : Sofic entropy is defined in terms of certain asymptotic geometric features of the sequences of metric spaces`Ω pO, σ n q, d pVnq˘.
Specifically, it is the quantity where cov δ is the δ-covering number of the given metric space and O runs over all weak˚-neighbourhoods of µ. The original motivation for studying sofic entropy was the classification of Bernoulli systems: see [3,12]. It has quickly found numerous other applications, and suggests many directions for further research. One of these is to look for other properties of the metric spaces pΩpO, σ n q, d pVnq q that carry some dynamical information about µ, and might give rise to other invariants.
The present paper focuses on one such property, which we refer to as 'connected model spaces rel Σ'. It is an approximate kind of connectedness for the spaces ΩpO, σ n q. In case X is finite, these metric spaces are discrete, so we do not ask for their connectedness as in classical point-set topology. Rather, we consider whether connectedness holds in a suitable asymptotic sense as O becomes smaller and n ÝÑ 8. The precise notion we need is given in Definition 2.5.
Once that definition has been made, we can show that the property of connected model spaces rel Σ is an isomorphism-invariant. In fact we prove more than this. Some factor maps do not preserve the property of connected model spaces, but we introduce a class of factor maps which do, and show that they include all isomorphisms.
Given a factor map Φ : pX G , µ, Sq ÝÑ pY G , ν, Sq, and also compact metrics d X on X and d Y on Y which generate their σ-algebras, the developments in [1] include the construction of suitable 'approximating maps' from X Vn to Y Vn . These send good models for µ to good models for ν, up to various errors that must be carefully controlled in terms of the metrics d pVnq X and d pVnq Y . Put roughly, we say that Φ is 'model-surjective rel Σ' if all good models for pY G , ν, Sq are close to images of good models for pX G , µ, Sq under these approximating maps. This is made precise in Definition 3.1.
Having introduced this special property of factor maps, we will show that it is preserved by composition, and that all isomorphisms have this property. Then we prove the following.
Theorem A Let Φ be a factor map as above. If pX G , µ, S, d X q has connected model spaces rel Σ and Φ is model-surjective rel Σ, then pY G , ν, S, d Y q also has connected model spaces rel Σ.
In particular, having connected model spaces rel Σ is a property only of the shift-system pX G , µ, Sq, not depending on the choice of d X , and is an isomorphisminvariant.
Having proved Theorem A, the property of having connected model spaces rel Σ can be extended unambiguously to any G-system, by picking an isomorphism from it to G-process.
Although factor maps need not preserve the connectedness of model spaces rel Σ, it turns out inverse limits do.
Theorem B If all members of an inverse sequence of G-systems have connected model spaces rel Σ, then so does the inverse limit.
The rest of the paper is given to some examples of connected and non-connected model spaces. In the first place, we have the following.
Theorem C For any sofic group G and sofic approximation Σ, Bernoulli systems over G have connected model spaces rel Σ.
On the other hand, for some groups G and sofic approximations Σ there are factors of Bernoulli shifts that do not have connected model spaces (and so, in particular, not all factor maps are model-surjective). The last section of the paper is given to a family of such examples.
Let m be the Haar probability measure on the circle T, and consider the factor map of G-systems pT G , mˆG, Sq ÝÑ pX, µ, T q (1) defined by forming the quotient by the diagonal subgroup of T G : that is, . . , θ, θ, . . .q : θ P Tu, µ is the Haar measure on this quotient group, and T is the quotient action.

Theorem D If G is a residually finite group with Kazhdan's property (T), then
G has a sofic approximation Σ relative to which the factor system pX, µ, T q constructed above does not have connected model spaces.
In particular, this implies that pX, µ, T q is not Bernoulli. These examples form a special case of a construction of non-Bernoulli factors of Bernoulli shifts due to Popa [18], so for general G we refer to pX, µ, T q as the Popa factor of pT G , mˆG, Sq. These factors show a striking difference between ergodic theory for amenable and non-amenable groups: among the former, factors of Bernoulli shifts are always still Bernoulli [17]. Popa's proof in [18] gives examples for a considerably larger range of groups G than those in Theorem D. It rests on calculations of the cohomology of these actions, generalizing work of Popa and Sasyk [19]. Our proof of Theorem D also uses some (more elementary) cohomology theory, and it covers fewer examples, but it gives a geometric interpretation to the non-Bernoullicity of these factors.
As pointed out to me by Brandon Seward, combining Theorems B, C and D immediately gives the following. This contrasts with actions of amenable groups, among which inverse limits of Bernoulli systems, like factors, are also always still Bernoulli [17].
In the reverse direction, there are many groups which admit inverse limits of Bernoulli shifts which are not factors of Bernoulli shifts. Indeed, if G has a nonamenable free subgroup, then Bowen has shown [4] that every Bernoulli shift over G factors onto every other Bernoulli shift (this property may in fact hold for all non-amenable G: see [5, Corollary 1.6] for partial progress). One may therefore form an inverse sequencë¨¨Ý of Bernoulli systems in which each A n is a finite alphabet and the Shannon entropies Hpp n q have finite sum. Now the inverse limit of this sequence has generating partitions of arbitrarily small Shannon entropy, by joining the generating partitions of all systems that are sufficiently high up the sequence. As a result, that inverse limit has Rokhlin entropy zero, and hence also sofic entropy at most zero [21,22]. On the other hand, every factor of a Bernoulli shift has positive sofic entropy [11]. Our tools can also be used to prove another strengthening of Theorem D. Recall that a factor map Φ : pX, µ, T q ÝÑ pY, ν, Sq of G-systems is complemented if there is another factor map Ψ : pX, µ, T q ÝÑ pZ, θ, Rq such that the combined map pΦ, Ψq is a measure-theoretic isomorphism pX, µ, T q -ÝÑ pYˆZ, νˆθ, SˆRq.
In this case pY, ν, Sq is a complemented factor of pX, µ, T q.
The Popa factors give examples of factor maps of Bernoulli shifts that are not model-surjective. However, it turns out that all complemented factor maps of Bernoulli shifts (and actually of a much more general class of systems) are modelsurjective: see Theorem 6.8. Combined with Theorems A and C, this shows the following.
Corollary D 2 If G is as in Theorem D then the Popa factor pX, µ, T q is not a complemented factor of any Bernoulli shift.
This corollary was also suggested to me by Brandon Seward. It can also be proved using Popa's cohomological approach, as has been shown to me by Yongle Jiang. The main new ingredient is the fact that, if pX, µ, T q is a complemented factor of any Bernoulli shift pX G , νˆG, Sq, then the resulting inclusion homomorphism from the degree-one, T-valued cohomology of the former into the latter is injective. Similar arguments can be found in Jiang's recent paper [9]. Finally, the above discussion suggests the following. Question 1.1. Is it true that for any group G, any complemented factor of a Bernoulli system is Bernoulli? ⊳ I do not know that the answer is Yes for any non-amenable group.

Model spaces and maps between them
This part of the paper follows closely the approach to sofic entropy developed in [1,Part I], and mostly uses the same notation.
We use 'big-O' and 'little-o' notation without further comment. Among real numbers, we sometimes write 'a « ε b' in place of '|a´b| ă ε'.
If pX, d X q and pY, d Y q are two metric spaces, ε ą 0 and L ă 8, then a map A map is ε-almost Lipschitz if this holds for some L.
We next recall some of the basic definitions and results from [1, Part I]. They are mostly small modifications to [3,14]. Suppose that pX G , µ, S, dq is a metric G-process, meaning that pX , dq is a compact metric space, S is the right-shift action of G on X G , and µ P ProbpX G q is S-invariant.
Given a finite set V and a map σ : G ÝÑ SympV q, and also v P V and x P X V , we define the pullback name of x at v by Π σ v pxq :" px σ g pvq q gPG P X G . In terms of these, the empirical distribution of x is For any w˚-neighbourhood O of µ in ProbpX G q, the O-good models for µ are the elements of ΩpO, σ n q :" tx : P σ x P Ou. Motivation for these concepts and their use in defining sofic entropy can be found in [1,Subsection 3.1].
Given any map ψ : X G ÝÑ Y and also a map σ : G ÝÑ SympV q for some finite set V , we define the associated map ψ σ : X V ÝÑ Y V by ψ σ pxq :"`ψpΠ σ v pxqq˘v PV , as in [1,Subsection 4.2]. Now let Φ " ϕ G : pX G , µ, S, d X q ÝÑ pY G , ν, S, d Y q be a factor map of metric G-processes, where we use the same notation as in [1,Subsection 4.1]. As in that reference, an η-almost Lipschitz (or η-AL) approximation to ϕ rel pµ, d X , d Y q is a measurable map ψ : X G ÝÑ Y with the following properties.
ii) There is a finite D Ď G such that ψ is D-local: that is, it depends only on coordinates in D.
iii) There is a D-local open subset U Ď X G such that µpUq ą 1´η and such that ψ|U is η-almost Lipschitz from d pDq X to d Y . These always exist for every η [1, Lemma 4.3]. In the applications below, we will more often consider sequences of such approximations: an almost Lipschitz approximating sequence for ϕ rel pµ, d X , d Y q is a sequence of maps ψ k : X G ÝÑ Y which are η k -AL approximations to ϕ rel pµ, d X , d Y q for some positive parameters η k ÝÑ 0. This situation is denoted by ψ k aL ÝÑ ϕ. Although we will be using such maps to study isomorphism-invariant properties of systems, the definitions above depend crucially on the choice of the compact metrics d X and d Y .
Later in this section, we will need some estimates from [1] concerning AL approximations and the corresponding maps on model spaces. Suitable versions are recalled in the following lemmas.
for all sufficiently large n.
For our later application of these results, it is convenient to combine them into a single assertion in terms of bases for the w˚-topologies around µ and ν. Since those topologies are metrizable, there exist such bases which are decreasing sequences of open neighbourhoods. We will frequently use this fact without further explanation.
• and a base of w˚-neighbourhoods N 1 Ě N 2 Ě . . . at ν such that for each k both of the following hold for all sufficiently large n: ψ σn k`Ω pO k , σ n q˘Ď ΩpN k , σ n q Proof. Let η k Ó 0 be a sequence such that ψ k is an η k -AL approximation to ϕ rel µ for every k, and set ε k :" 3η k for each k.
Let M 1 Ě M 2 Ě . . . be a base of w˚-neighbourhoods at ν with M 1 " ProbpY G q. By Lemma 2.1, for each j P N there are k 0 pjq P N and w˚-neighbourhoods O 1 j,k for each k ě k 0 pjq such that for every k ě k 0 pjq we have ψ σn k`Ω pO 1 j,k , σ n q˘Ď ΩpM j , σ n q for all sufficiently large n.
We may take k 0 p1q " 1 because M 1 " ProbpY G q, and we may also assume that k 0 pjq ÝÑ 8 as j ÝÑ 8. Now let By forming running intersections we may also assume that O 1 1 Ě O 1 2 Ě . . . . For each k, letting j be the largest integer for which k 0 pjq ď k, it follows that ψ σn k`Ω pO 1 k , σ n q˘Ď ψ σn k`Ω pO 1 j,k , σ n q˘, and this is contained in ΩpM j , σ n q " ΩpN k , σ n q for all sufficiently large n.
On the other hand, Lemma 2.2 gives values K k ă 8 and w˚-neighbourhoods O 2 k of µ such that ψ σn k |ΩpO 2 k , σ n q is ε k -almost K k -Lipschitz for all sufficiently large n, for every k ě 1. Once again we may assume that O 2 The deduction of this corollary from Lemmas 2.1 and 2.2 is similar to the deduction of 'sequence versions' of those lemmas in [1,Subsection 4.3]. But it seems easier to make this deduction from scratch here, rather than adapting the corollaries in that subsection. Corollary 2.3 has a further consequence that is worth recording by itself. ψ σn k`Ω pO k , σ n q˘Ď ΩpN k , σ n q for all sufficiently large n.
Since P σn xn weakÝ Ñ µ, for every k there is an Npkq such that x n P ΩpO k , σ n q for all n ě Npkq. Provided the sequence k 1 ď k 2 ď . . . grows sufficiently slowly, it follows that x n P ΩpO kn , σ n q for all sufficiently large n, and also that ψ σn kn`Ω pO kn , σ n q˘Ď ΩpN kn , σ n q for all sufficiently large n. These together imply that ψ σn kn px n q P ΩpN kn , σ n q for all sufficiently large n, and so their empirical distributions converge to ν, since N 1 Ě N 2 Ě . . . is a base for the w˚-topology at ν.

Connected model spaces
Let pY, d Y q be a metric space, let x, y P Y , and let δ ą 0. A δ-path from x to y is a finite sequence x " x 0 , x 1 , . . . , x ℓ " y in Y such that d Y px i , x i`1 q ă δ for every i P t0, . . . , ℓ´1u. The integer ℓ is the length of this δ-path. If A Ď Y and δ ą 0, then A is δ-connected (according to d Y ) if for any x, y P A there is a δ-path from x to y contained in A.
We are now ready to define our new property of metric G-processes: Definition 2.5 (Connected model spaces rel Σ). Let pX G , µ, S, dq be a metric Gprocess. It has connected model spaces rel Σ if the following holds: If n 1 ă n 2 ă . . . , and x i , y i P X Vn i are two sequences satisfying then there are a sequence δ i Ó 0 and a sequence of δ i -paths for all sufficiently large i.
This definition is made more complicated by the allowance of an arbitrary subsequence n 1 ă n 2 ă . . . . This is because of cases in which there are some other subsequence n 1 1 ă n 1 2 ă . . . and a w˚-neighbourhood O such that ΩpO, σ n 1 i q " H for all i. Such cases should still be called 'connected' if any two sufficiently good models, for a sufficiently large value of n, can be joined by a δ-path consisting of fairly good models. This requirement simply ignores any values n 1 for which σ n 1 admits no good models at all. However, if there is a subsequence n 1 1 ă n 1 2 ă . . . as above, then there can be no sequences x n , y n P X Vn defined for all integers n which satisfy P σn xn , P σn yn weakÝ Ñ µ, and we must pass to a subsequence which eventually avoids the n 1 i s. This will be made clearer by the proof of Proposition 2.6 below.
Definition 2.5 can be re-written more directly in terms of connectedness properties of the model spaces ΩpO, σ n q. At first sight, it seems similar to requiring that these spaces are δ-connected for all sufficiently large n, but that impression is not quite correct. In order to fix it, we need a slightly more complicated notion.
If pY, d Y q is any metric space and A Ď B Ď Y is a nested pair of subsets, then the pair pA, Bq is relatively δ-connected (according to d Y ) if for any x, y P A there is a δ-path from x to y contained in B. Proposition 2.6. If pX G , µ, S, dq is a metric G-process, then the following are equivalent.
1. It has connected model spaces rel Σ.

For every δ ą 0 and every w˚-neighbourhood
ΩpO 1 , σ n q, ΩpO, σ n qȋ s relatively δ-connected according to d pVnq for all sufficiently large n.
These properties are both implied by the following.
3. For every δ ą 0, µ has a base of w˚-neighbourhoods N with the property that the set ΩpN , σ n q is δ-connected for all sufficiently large n.
Proof. (1. ùñ 2.) If property 2 does not hold, then there are some δ and O for which it fails. Let O 1 Ě O 2 Ě¨¨¨be a base for the w˚-topology at µ such that O i Ď O for every i. By the failure of property 2, there are integers n 1 ă n 2 ă . . . and pairs tx i , y i u Ď ΩpO i , σ n i q for every i which cannot be connected by δ-paths that stay inside ΩpO, σ n i q. This prevents the existence of δ-paths from x i to y i satisfying (3) in Definition 2.5.
. be a base for the w˚-topology at µ.
For each j, property 2 gives a sub-neighbourhood O 1 j Ď O j such that the pair ΩpO 1 j , σ n q, ΩpO j , σ n q˘ (4) is relatively 2´j-connected (according to d pVnq ) for all sufficiently large n.
such that x i , y i P ΩpO 1 j , σ n i q for all i ě i j . Now the relative 2´j-connectedness of the pair (4) implies that for each i ě i j there is a 2´j-path These paths verify Definition 2.5 with (3. ùñ 2.) For any O, property 3 gives a sub-neighbourhood N Ď O for which ΩpN , σ n q is δ-connected for all sufficiently large n, and this implies that the pair`Ω pN , σ n q, ΩpO, σ n qȋ s relatively δ-connected.
In all the examples of connected model spaces that I know, one actually has property 3 above, which is easier to verify. But I do not see a proof that these are equivalent, and I also do not know whether property 3 is isomorphism-invariant.
Remark. Definition 2.5 has a natural modification as follows. Let us say that pX G , µ, S, dq has uniformly connected model spaces rel Σ if there is a function ℓ : p0, 1q ÝÑ N for which following holds: If n 1 ă n 2 ă . . . , and x i , y i P X Vn i are two sequences satisfying then for every δ ą 0 and all sufficiently large i there are δ-paths That is, one requires that the lengths of the δ-paths depend only on δ, not on n. It is easily shown that this variant is formally stronger than Definition 2.5 In the case of our principal examples, Bernoulli shifts, the proof below actually shows that this stronger property holds. I do not know of any examples that have connected but not uniformly connnected model spaces. ⊳

Model-surjective factor maps and Theorem A
As promised in the Introduction, for some groups G the Popa factor does not preserve the connectedness of model spaces. We now introduce a special kind of factor map which does always preserve this property, and show that all isomorphisms are factor maps of this kind. They are defined in terms of good models for the associated graphical joinings, but we will prove an equivalent characterization in terms of AL approximations to the factor map. We finish the subsection by proving Theorem A. Later we show that complemented factor maps of Bernoulli shifts are also of this kind, which leads to the proof of Corollary D 2 .
be a factor map of metric G-processes, and let λ " ż X G δ px,Φpxqq µpdxq be the associated graphical joining of µ and ν. Also, let d be the Hamming average of the metrics d X and d Y on XˆY.

Definition 3.1 (Model-surjective factor maps rel Σ). This factor map
where this last convergence refers to the w˚-topology arising from the product topology on X GˆY G .
Clearly the sequence x i obtained above must also satisfy P σn The idea behind Definition 3.1 is that, if the joint empirical distribution converges to λ, then this forces x i to 'resemble' a pre-image of y i under some model-space approximation to the factor map Φ.
The need to consider a sofic sub-approximation pσ n i q iě1 of Σ is similar to the case of Definition 2.5.
Beware that at this point, Definition 3.1 requires a particular choice of the metrics d X and d Y . Once we have shown that isomorphisms are model-surjective (Proposition 3.6 below), it will follow that Definition 3.1 actually depends only on the measure theoretic structure of Φ as a factor map from pX G , µ, Sq to pY G , ν, Sq, and moreover that it can be extended unambiguously to factor maps of general Gsystems (Corollary 3.7).
First we need a simple lemma and corollary which relate the metrics d pVnq and certain empirical distributions.

Lemma 3.2.
Let Σ be as before, and let pX , dq be a compact metric space. For any two sequences x n , z n P X Vn , the following are equivalent: 2. in the w˚topology, any subsequential limit of the sequence In case P σn xn w˚-converges to some µ P ProbpX G q, either of the above conditions implies that P σn zn w˚-converges to the same limit. Proof. On X GˆX G , consider the function It is continuous, and a simple calculation gives ż F dP σn pxn,znq " d pVnq px n , y n q.
Assuming condition (1), it follows that any subsequential limit λ " lim jÝÑ8 P σn j pxn j ,zn j q must satisfy ş F dλ " 0, hence be supported on the set tpx, x 1 q : x e " x 1 e u. Since λ is S-invariant (see [1, Lemma 3.2]), it must be supported on the diagonal.
On the other hand, if (1) fails, then there are some δ ą 0 and some subsequence n 1 ă n 2 ă . . . for which d pVn j q px n j , z n j q ÝÑ δ as j ÝÑ 8.
By the sequential compactness of the w˚-topology on ProbpX GˆX G q, there is a further subsequence for which the empirical distributions converge to some λ. This λ must then satisfy ş F dλ " δ ą 0, and so cannot be supported on the diagonal.
Finally, any measure supported on the diagonal must have equal first and second marginals, so condition (2) clearly implies the last part of the lemma.
be a factor map of metric G-processes, and let λ " be the resulting graphical joining of these two systems. Suppose that x n P X Vn and y n , w n P Y Vn are sequence such that Proof. For each n, let θ n :" P σn pxn,yn,wnq P ProbppXˆYˆYq G q, and let θ :" lim jÝÑ8 θ n j be any w˚-subsequential limit of these measures. The projections of θ n onto the two copies of pXˆYq G are P σn pxn,ynq and P σn pxn,wnq , so we know that the limiting measure θ has both of these projections equal to λ. Hence θ is supported on tpx, y, y 1 q : x P X G , y " ϕ G pxq " y 1 u.
Therefore the limit of P σn pyn j ,wn j q is supported on the diagonal in pYˆYq G , and Lemma 3.2 completes the proof. Now we can begin the study of model-surjectivity. Graphical joinings give the easiest way to define this property, but it is useful to have a more 'functional' characterization. This can be given in terms of AL approximations to the factor map.

Lemma 3.4.
For Φ " ϕ G as above, the following are equivalent.

Suppose that
Proof. Let ψ k aL ÝÑ ϕ be an AL approximating sequence rel pµ, d X , d Y q. Also, let ξ : X G ÝÑ X be the projection onto the teu-coordinate, so ξ G " id X G . An easy check (or see [1,Corollary 4.7]) gives that where d is the Hamming average of d X and d Y on XˆY, and pξ, ϕq denotes the map X G ÝÑ XˆY : x Þ Ñ pξpxq, ϕpxqq.
Let n 1 ă n 2 ă . . . , and let y i P Y Vn i be a sequence whose empirical distributions tend to ν. By re-labeling the sofic sub-approximation pσ n i q iě1 if necessary, we may assume that n i " i for all i, and hence write y n as an element of Y Vn .
The result follows because Lemma 3.2 (respectively Corollary 3.3) shows that d pVnq`p x n , y n q, px n , ψ σn kn px n qq˘" are both model-surjective rel Σ, then so is their composition.
Proof. As in the proof above, after passing to a sofic sub-approximation we may assume that z n P Z Vn is a sequence whose empirical distributions tend to θ. By the two assumed instances of model-surjectivity rel Σ, we may find first a sequence y n P Y Vn and then a sequence x n P X Vn such that where λ and r λ are the graphical joinings associated to Φ and r Φ respectively. Now let λ 1 " lim jÝÑ8 P σn j pxn j ,yn j ,zn j q be any subsequential limit of the triple empirical distributions. Since its projection onto the first two coordinates must be λ and its projection onto the second two coordinates must be r λ, it is supported on the set `x , Φpxq, r ΦpΦpxqq˘: x P X G ( , and hence it must be the full graphical joining associated to pΦ, r Φ˝Φq. Therefore any subsequence of P σn pxn,znq converges to the graphical joining associated to r Φ˝Φ, as required.
Proposition 3.6. If Φ " ϕ G is an isomorphism, then it is model-surjective for any sofic approximation Σ.
Proof. Let r ϕ be such that Φ´1 " r ϕ G , let ξ : Y G ÝÑ Y be the projection to the teu-indexed coordinate, and let r ψ m aL ÝÑ r ϕ rel pν, d Y , d X q. Let λ be the graphical joining of Φ. Then we also have (see again [1,Corollary 4.7] for a careful proof of this).
As in the proofs above, to show the model-surjectivity of Φ we may pass to a sofic sub-approximation, and so suppose that y n P ProbpY Vn q is a sequence whose empirical distributions converge to ν. Then (6) and Lemma 2.1 give that P σn p r ψ σn mn pynq,ynq weakÝ Ñ λ provided pm n q ně1 grows sufficiently slowly. Letting x n :" r ψ σn mn py n q, this completes the proof. The first important consequence of Propositions 3.5 and 3.6 is the following.
is a commutative square of factor maps in which the horizontal arrows are isomorphisms and Φ is model-surjective rel Σ, then Ψ is also model-surjective rel Σ. In particular, model-surjectivity rel Σ in Definition 3.1 is independent of the choice of generating metrics d X and d Y .
We now have all the ingredients needed to prove Theorem A.
Proof of Theorem A. Let n 1 ă n 2 ă . . . , and let y i , w i P Y Vn i be sequences such that P σn i We must show that these pairs may be connected by op1q-paths consisting of good models, as in Definition 2.5. By passing to the sofic sub-approximation pσ n i q iě1 , we may relabel all these sequences and so assume that i " n i for all i, and hence write the index as n itself. Now let x n , z n P X Vn be the sequences given by the model-surjectivity of Φ applied to the sequences y n and w n , respectively. Since pX G , µ, S, d X q has connected model space rel Σ, there are parameters δ n Ó 0 and a sequence of δ npaths x n " x n,0 , x n,1 , . . . , x n,ℓn " z n which are eventually contained in ΩpO, σ n q for any w˚-neighbourhood O of µ.
Also, let ψ k aL ÝÑ ϕ rel pµ, d X , d Y q. Corollary 2.3 gives parameters ε k Ó 0 and K k ă 8, a sequence of w˚-neighbourhoods O 1 Ě O 2 Ě . . . of µ, and a base of w˚-neighbourhoods N 1 Ě N 2 Ě . . . at ν, such that for every k we have both ψ σn k |ΩpO k , σ n q is ε k -almost K k -Lipschitz for all sufficiently large n (7) and ψ σn k`Ω pO k , σ n q˘Ď ΩpN k , σ n q for all sufficiently large n.
Now choose a sequence k 1 ď k 2 ď . . . growing so slowly that all of the following hold: i) we have ε kn`Kkn δ n ÝÑ 0 as n ÝÑ 8 (this is possible because ε k ÝÑ 0 and δ n ÝÑ 0); ii) we have tx n,0 , x n,1 , . . . , x n,ℓn u Ď ΩpO kn , σ n q for all sufficiently large n; iii) we have iv) we have ψ σn kn |ΩpO kn , σ n q is ε kn -almost K kn -Lipschitz for all sufficiently large n, as is possible by (7) v) we have ψ σn kn`Ω pO kn , σ n q˘Ď ΩpN kn , σ n q for all sufficiently large n, as is possible by (8).
By properties (ii) and (iv), the sequence y n,0 , y n,1 , . . . , y n,ℓn is a pε kn`Kkn δ n q-path according to d pVnq Y for all sufficiently large n. Combined with properties (i) and (iii), this shows that y n , y n,0 , y n,1 , . . . , y n,ℓn , w n is a δ 1 n -path from y n to w n for some sequence of parameters δ 1 n Ó 0. Finally, properties (ii) and (v) imply that these image paths are eventually contained in ΩpN , σ n q for every w˚-neighbourhood N of ν, because N 1 Ě N 2 Ě . . . is a base at ν.

Connected model spaces for inverse limits
This section gives the proof of Theorem B.
Proof of Theorem B. Up to isomorphism, the setting of this theorem may be represented as follows. Let pX G i , µ i , Sq, i " 1, 2, . . . be an infinite sequence of G-processes, and let be a joining of all of them. Let π k : ś iě1 X i ÝÑ ś k i"1 X i be the coordinate projection for each k, and let λ k :" π G k˚λ . We assume that the G-systeḿ as connected model spaces rel Σ for each k, and must prove the same for the infinite joining λ. This is easiest using the reformulation in condition 2 of Proposition 2.6.
For each i, let d X i be a compact generating metric of diameter at most 1 for the space X i . Define metrics d k on ś k i"1 X i and d on so these generate the compact product topologies on their respective spaces. Now suppose that O is a w˚-neighbourhood of λ. By shrinking it if necessary, we may assume that it has the form θ : π G k˚θ P O 1 ( for some k P N and some w˚-neighbourhood O 1 of λ k , since sets of this form are a base of w˚-neighbourhoods around λ. Under this assumption, we obtain also ΩpO, σ n q " Equation (9) has an obvious analog for O 1 and O 1 1 . Combining these, it follows that the pair`Ω pO 1 , σ n q, ΩpO, σ n qȋ s also relatively δ-connected according to d pVnq for all sufficiently large n. This verifies condition 2 in Proposition 2.6 for the process defined by λ.

Connected model spaces for Bernoulli systems
This subsection proves Theorem C. We actually prove the slightly stronger property (3) from Proposition 2.6. Let pX , dq be a compact metric space of diameter at most 1, and let ν P ProbpX q. This section makes several simple appeals to the phenomenon of measure concentration for product measures and Hamming metrics: see, for instance, [16] for a dedicated exposition and [8, Chapter 3 1 2 ] for a geometrically-flavoured overview. The specific result that we need is the following: see [16, Corollary 1.17]. Next we identify certain special neighbourhoods of the product measure νˆG. Let F be a finite subset of G, and for each D Ď F define an operator E D on CpX F q as follows. First, if F " tg 1 , . . . , g m u and D " F ztg j u for some j ď m, then E D f pxq :" ż X f px g 1 , . . . , x g j´1 , y, x g j`1 , . . . , x gm q νpdyq.
In general, if D " F ztg j 1 , . . . , g j k u, then where the order of this composition is unimportant. This E D is the operator of conditional expectation with respect to νˆF onto the σ-algebra of Borel subsets of X F that depend only on coordinates in D. It follows that Now suppose that F Ď CpX G q is a family of F -local functions. Let us call it hereditary if every member of F is 1-Lipschitz according to d pF q X and if It is easily checked that each E D preserves the property of being 1-Lipschitz with respect to d pF q X . A hereditary neighbourhood of νˆG is a w˚-neighbourhood of the form for some finite F Ď G, some ε ą 0, and some finite hereditary family F of F -local continuous functions.

Proposition 5.2.
If O is a hereditary neighbourhood of νˆG and δ ą 0, then the set ΩpO, σ n q is δ-connected according to d pVnq for all sufficiently large n.
We will see the importance of assuming that O is hereditary during the course of the proof.
Proof of Theorem C from Proposition 5.2. Hereditary neighbourhoods form a base for the w˚-topology at µ. Therefore Proposition 5.2 verifies condition 3 in Proposition 2.6, which implies connected model spaces.
Proposition 5.2 will be proved by showing how any pair of points in ΩpO, σ n q may be connected by a 'random' δ-path, provided n is sufficiently large.
We insert randomness into the proof as follows. Fix κ P p0, 1q, to be specified later. For each n, let pξ n,t q tě0 be a discrete-time random walk on X Vn with the following transition probabilities: where we write ξ n,t " pξ n,t,v q vPVn . Thus, ξ n,t`1 is obtained from ξ n,t by considering each coordinate in X Vn independently, and either re-sampling it from the distribution ν with probability 1´κ, or leaving it unchanged with probability κ.
Let P x n be the law of pξ n,t q tě0 on X VnˆX Vnˆ¨¨¨c onditioned on starting from ξ n,0 " x, and let E x n denote expectation with respect to P x n . We will find a δ-path between two good models x and y by starting a copy of this random walk at each of x and y, and showing that after a certain bounded time the following hold: (i) these random walks have probably stayed inside the set of good models, (ii) they have probably taken only steps smaller than δ in the Hamming distance, and (iii) they can be coupled in such a way that with high probability they end up close to each other.
We break the necessary estimates into three separate lemmas. Proof. Let O be as in (11) for some ε ą 0 and some hereditary family F of F -local functions.
Since F is finite, it suffices to show that for any one f P F we have We break this estimate into two further steps.
Step 1. Observe that Since F is finite and Σ is a sofic approximation, as n ÝÑ 8 it holds with high probability in the choice of v P V n that the points σ g n pvq for g P F are distinct.
For such v, iterating the formula (12) gives Let D be a random subset of F which contains each element of F independently with probability κ t . Let P 1 be its law and let E 1 be expectation with respect to P 1 . Then the above leads to where the op1q-correction results from those few vertices v which fail the requirement (13). ThereforeˇˇˇE x n ż f dP σn ξ n,t´ż f dνˆGˇď where the equality of second and third lines uses (10), and the last equality uses that E H f is constant and equal to ş f dνˆF . Since x P ΩpO, σ n q with O as in (11), the last line above is strictly bounded by where the op1q-correction depends only on the sofic approximation Σ and on }f } 8 . Therefore this bound is strictly less than, say, p1´1 2 p1´κ t q |F | qε for all sufficiently large n.
Step 2. On the other hand, we may regard the quantity ş f dP σn ξ n,t as a random variable on the probability spacè X Vn , P x n tξ n,t P¨u˘.
Using the initial condition ξ n,0 " x and the transition probabilities (12), the probability measure here is equal to ą vPVn`κ which is a product measure on X Vn . By the definition of the empirical distribution, and recalling that f is F -local and 1-Lipschitz according to d pF q X , this random variable is an |F |-Lipschitz function on this product space. Therefore Proposition 5.1 gives some β ą 0, depending only on the ratio p1´κ t q |F | ε{|F |, such that P x n !ˇˇˇż f dP σn ξ n,t´E x n ż f dP σn ξ n,tˇě Thus this probability tends to 0 as n ÝÑ 8 uniformly in x.
Combining the estimates from Steps 1 and 2 completes the proof.
Proof. By time-homogeneity, it suffices to prove this when t " 0.
Since diampX , dq ď 1, we always have This is an average of indicator functions of independent events, all of them having probability at most 1´κ ă δ. Therefore another appeal to Proposition 5.1 (or just the special case of a Chernoff bound) gives a β ą 0 for which P x n d pVnq px, ξ n,1 q ě δ ( " Ope´β |Vn| q.
Since the right-hand bound is independent of x, this completes the proof.
Lemma 5.5. If s P N is so large that κ s ă δ{4, then the following holds. For any n P N and any x, y P X Vn , the distributions P x n pξ n,0 , . . . , ξ n,s q P¨( and P y n pζ n,0 , . . . , ζ n,s q P¨( of the random walks started at x and y up to the finite time-horizon s have a coupling Q such that Proof. By an induction on s using (12), we have P x n tξ n,s P¨u " ą vPVn`κ s δ xv`p 1´κ s qν˘, and similarly for P y n tζ n,s P¨u. Consider a random triple pα, ξ, ζq of elements of X Vn with law constructed as follows. First, choose α from the law νˆV n . Then, for each v P V n independently, choose two random bits η v , ω v P t0, 1u independently, each equal to 1 with probability κ s . Finally, for each v P V n , set Letting λ be the joint distribution of pα, ξ, ζq, it follows that λtξ P¨u " P x n tξ n,s P¨u, λtζ P¨u " P y n tζ n,s P¨u, and ż d pVnq pζ, ξq dλ ď 2κ s ă δ{2.
Thus, under λ, the pair of random variables pξ, ζq are a coupling of pξ n,s , ζ n,s q under which the probability of the event td pVnq pξ n,x , ζ n,s q ă δu is greater that 1{2, by Chebyshev's Inequality. Now we can choose any extension of this to a coupling Q of the whole random trajectories pξ n,0 , . . . , ξ n,s q and pζ n,0 , . . . , ζ n,s q: for instance, we can couple them relatively independently over the given coupling of the end-states ξ n,s and ζ n,s .

Remark. Using Chernoff's Inequality for the random sum
ř v pη v`ωv q, one can actually improve (14) to a lower bound of the form 1´Ope´β |Vn| q, but we will not need this. ⊳

Proof of Proposition 5.2.
Let O be a hereditary w˚-neighbourhood of νˆG and let δ P p0, 1q. Choose some κ P p1´δ, 1q, and then choose s P N so that κ s ă δ{4.
Suppose that x, y P ΩpO, σ n q, and let Q be a coupling of two trajectories of the random walk up to time s, one starting from x and the other from y, as constructed in Lemma 5.5. Then the conjunction of the bounds (15), (16) and (14) shows that the event !
Remark. Since the choice of s in the above proof depends only on δ, it actually shows that Bernoulli shifts have uniformly connected model spaces rel Σ, as in the remark at the end of Subsection 2.2. ⊳

Actions and cocycles for property-(T) groups
Let G be a finitely generated group. Let S be a finite and symmetric generating set, and let R be the set of all the corresponding relations in the free group on S (including concatenations or conjugates of other relations). Recall that G has Kazhdan's property (T) if there is a c ą 0 for which the following holds: whenever π : G ñ V is a unitary representation, if there is some v P V such that }v} " 1 and then π has a nontrivial invariant vector. The value of c depends on the choice of S, but its existence does not. See, for instance, [2]. Now suppose that pX, µ, T q is a G-system and that K ď T is a closed subgroup. Let Upµ, Kq be the set of measurable functions X ÝÑ K modulo agreement µ-a.e. This is a group under pointwise addition, and is naturally equipped with the topology of convergence in probability. That topology is Polish, with a suitable metric given by the group-norm }f } µ :" where |¨| is the quotient group-norm on T of the usual absolute value on R. The action T : G ñ pX, µq induces an action of G on Upµ, Kq. We will need to work with the cohomology of this action in degree 1, which is conveniently expressed in terms of the generators S and relations R. Firstly, a K-valued 1-cochain is an equivalence class modulo µ of measurable functions α : SˆX ÝÑ K, or equivalently an element of Upµ, Kq S . For a 1cochain α, we set }α} µ,S :" ÿ sPS }αps,¨q} µ .
These form a further subgroup B 1 pT, µ, Kq ď Z 1 pT, µ, Kq, not necessarily closed, and the quotient of these groups is the first cohomology group H 1 pT, µ, Kq. The next result is due to Schmidt [20,Theorem 3.4] and independently to Zimmer [23,Theorem 2.11]. We include a proof in order to show that the relevant constants do not depend on the action T , or on the choice of the closed subgroup K of T. Theorem 6.1. If G has property (T), then there is some r ą 0 with the following property. Let pX, µ, T q be an ergodic G-system. For any closed subgroup K ď T and any α P Z 1 pT, µ, Kq, we have }α} µ,S ď r ùñ α P B 1 pT, µ, Kq.
Proof. Step 1. First suppose that K " T.
Let c be the constant in the definition of property (T), let r :" c 2 {4π, and consider α P Z 1 pT, µ, Tq with }α} µ,S ď r. Let π : G ñ L 2 pµq be the Koopman representation of T twisted by α: pπ g f qpxq :" e 2πiαpg´1,xq¨f pT g´1 xq. This is a well-defined G-action because α satisfies the defining equations for a 1-cocycle. Now observe that where 1 X is the constant function 1 on X. Therefore property (T) gives some f P L 2 pµq such that }f } L 2 pµq " 1 and π g f " f for all g P G.
This fixed-point equation implies that |f | is T -invariant, hence µ-a.s. constant by ergodicity. Therefore f is actually S 1 -valued, and the fixed-point equation reads e 2πiαpg,xq " f pT g xqf pxq @g P G.
Taking arguments, this asserts that α P B 1 pT, µ, Tq, as required.
Since α is actually K-valued, this implies that the coset β 0 pxq`K is T -invariant, and hence a.s. constant by ergodicity. Let c`K be that coset, and let βpxq :" β 0 pxq´c. Then β takes values in K almost surely, and still satisfies αpg, xq " βpT g xq´βpxq for µ-a.e. x @g P G.
The set of these will be denoted Z 1 F,ε pT, µ, Kq. For these we have the following roughened version of Theorem 6.1. Theorem 6.2. Suppose G " xS | Ry has property (T), and let r ą 0 be as Theorem 6.1. Then for every r 1 ą 0 there are ε ą 0 and finite F Ď R such that the following holds. If pX, µ, T q is an ergodic G-system, and α P Z 1 F,ε pT, µ, Kq satisfies }α} µ,S ď r, then there is some β P Upµ, Kq such that }α´dβ} µ,S ă r 1 . Proof.
Step 1. We first convert the result to an assertion about invariant measures on a fixed G-space. Let Y :" pK S q G " K SˆG , equipped with the G-action by coordinate right-shift, and define the canonical 1-cochain α 0 : SˆY ÝÑ K by α 0 ps, yq :" y s,e .
We will prove that the desired result is implied by the following: For every r 1 ą 0 there are ε ą 0 and finite F Ď G such that the following holds. If ν is an ergodic shift-invariant Borel probability on Y , and if }α 0 } ν,S ď r and }dα 0 } ν,F ă ε, then there is some β 0 P Upν, Kq such that }α 0´d β 0 } ν,S ă r 1 .
Indeed, suppose this result is known, and consider pX, µ, T q and α as in the statement of the theorem. Define ϕ : X ÝÑ Y : x Þ Ñ pαps, T g xqq sPS,gPG .
Step 2. The rest of the proof is by contradiction. Fix an increasing sequence pF i q iě1 of finite sets whose union is R, and suppose that one could find r 1 ą 0 and a sequence pν i q iě1 of ergodic shift-invariant Borel probabilities on Y such that }α 0 } ν i ,S ď r and }dα 0 } ν i ,F i ă 2´i (18) for all i, but also such that for all measurable functions β 0 : Y ÝÑ K and all i. By passing to a subsequence, we may assume that these measures ν i weak˚converge to a measure ν, which must still be shift-invariant. Since G has property (T), a theorem of Glasner and Weiss [7] gives that the set of ergodic measures is weak˚-closed among the shift-invariant probability measures on Y , so this ν is still ergodic. Now, since each α 0 ps,¨q is continuous on Y , the two parts of (18) give Therefore α 0 : SˆY ÝÑ K is an element of Z 1 pshift, ν, Kq with }α 0 } ν,S ď r, so Theorem 6.1 gives some β 1 P Upν, Kq such that α 0 " dβ 1 ν-a.s..
Since β 0 is continuous and ν i weakÝ Ñ ν, this implies that also for all sufficiently large i. This contradicts (19).
Since the bounds in Theorem 6.2 do not depend on the system pX, µ, T q, they are nontrivial even for actions on finite sets. In particular, suppose that H ă G is a finite-index subgroup, let V :" G{H, and let Γ :" pV, Eq be the directed Schreier graph resulting from the generating set S. Let G act on V by left-multiplication. Observe that SˆV is in canonical bijection with E, so a 1-cochain for this system may be interpreted as a map α : E ÝÑ K. For any F Ď R, let L F be the set of based loops in Γ that correspond to walking around a relation from the set F , starting from any vertex. We will identify such a loop by a pair pv, wq, where v is its starting vertex and w is the relation. For any α : E ÝÑ K, we can now define dα : L R ÝÑ K by dαpv, wq :" ℓ´1 ÿ where w " s ℓ s ℓ´1¨¨¨s1 . As before, a map α : E ÝÑ K is a 1-cocycle if and only if dα " 0: that is, it sums to zero around any loop in Γ which corresponds to a relation of the group.
Similarly to the case of general G-systems, let Z 1 pΓ, Kq be the set of 1cocycles α : E ÝÑ K, let B 1 pΓ, Kq " dpK V q be the subgroup of coboundaries, and for ε ą 0 and F Ď R let Z 1 ε,F pΓ, Kq be the subset of 1-cochains α which satisfy 1 |V | ÿ vPV |dαpv, wq| ă ε @w P F.
We also write d for the restriction of this metric to K S for any closed subgroup K of T. Since SˆV is canonically identified with E, the normalized Hamming sums d pV q may be regarded as metrics on T E . Now the translation of Theorems 6.1 and 6.2 into this setting implies the following. Proof. Suppose dβ P B 1 pΓ, Kq is such that d pV q pα, dβq ď r. It suffices to prove either part with α replaced by α´dβ, and so we may simplify the assumption to d pV q pα, 0q ď r. What remains is a special case of Theorem 6.2.
Remark. The relation between ε and r 1 in part 2 of this corollary is reminiscent of recent work on coboundary expansion for simplicial complexes by Kaufman, Kazhdan and Lubotzky [10]. They prove a similar inequality for K " Z{2Z and for a family of graphs Γ constructed from certain Bruhat-Tits buildings. Their family of examples is much more specialized than ours, but they obtain a more quantitative result: a linear dependence of ε on r 1 . They need this extra precision to deduce some other expansion properties of these complexes. It would be interesting to know whether one can be so precise under our more general conditions. ⊳

Proof of Theorem D
Now let G be an infinite, finitely generated, residually finite group having Kazhdan's property (T), and let e be its identity element. Let S and R be as in the previous subsection. Such a group G has a descending sequence G 1 ą G 2 ą . . . of finite-index subgroups such that Ş n G n " teu. For the proof of Theorem D we will need those subgroups to have two additional properties. Lemma 6.4. There is a descending sequence pG n q ně1 of finite-index subgroups as above such that (i) they all have nontrivial Abelianization, and (ii) the left-multiplication actions σ n : G ÝÑ SympG{G n q form a sofic approximation to G.
Proof. First, as is standard, there is a sequence G ą H 1 ą H 2 ą . . . of finiteindex normal subgroups converging to teu. For each n, consider the quotient homomorphism Since the target is a nontrivial group, it contains a nontrivial cyclic subgroup K n ď H n {H n`1 . Letting G n :" q´1 n pK n q, we obtain H n ě G n ą H n`1 . Let σ n be the left-multiplication action of G on G{G n . We can now deduce the required properties.
(i) By construction, K n is a quotient group of G n {rG n , G n s, so this latter is nontrivial for every n.
(ii) If left-multiplication by g P G has a fixed point in G{G n , then it has a fixed point in the further quotient G{H n . Since H n is normal in G, this requires that g P H n . Since the subgroups H n converge to teu, it follows that, for any g ‰ e, the permutation σ g n has no fixed points for all sufficiently large n. This implies that pσ n q ně1 is a sofic approximation.
Henceforth G, S and the sequence pG n q n given by the above lemma will be fixed. Let Γ n :" pV n , E n q be the sequence of directed Schreier graphs on the vertex sets V n " G{G n and with E n defined by the generating set S. These form a sofic approximation to G via the homomorphisms σ n in part (ii) of the above lemma. In this setting we often write g¨v in place of σ g n pvq for g P G and v P V n . Now let us return to the Popa factor map described in the Introduction. An isomorphic factor map whose target is a shift-system may be constructed as follows: Φ : T G ÝÑ T SˆG : pθ g q gPG Þ Ñ pθ sg´θg q sPS,gPG .
Since ϕ depends on only finitely many coordinates in T G , it is an η-AL approximation to itself for every η ą 0, and we may work directly with the maps ϕ σn on model spaces. This Φ is a homomorphism of compact Abelian groups. Since S generates G, the kernel of Φ is the diagonal subgroup of T G . The image of Φ is a compact subgroup of T SˆG , which we denote by Z. The image measure Φ˚mˆG must equal the Haar measure µ Z of Z. Therefore Φ is a factor map pT G , mˆG, Sq ÝÑ pT SˆG , µ Z , Sq which is equivalent to the Popa factor map from the Introduction.
To prove Theorem D, we will need a description of the spaces of good models for this factor system. This begins with the following, which is an immediate consequence of the definitions. Lemma 6.5. For any finite F Ď R and ε ą 0 there is a w˚-neighbourhood O of µ Z such that ΩpO, σ n q Ď Z 1 ε,F pΓ n , Tq @n ě 1.
If E Ď G and π E : T SˆG ÝÑ T SˆE is the coordinate projection, then π E is a group homomorphism, and the image measure pµ Z q E " pπ E q˚µ Z equals the Haar measure µ π E pZq . Lemma 6.6. Let E Ď G be finite, let C Ď G be a finite subset which is connected in CaypG, Sq and such that C Ě E Y SE, and let n be so large that, for any v P V n , the map C ÝÑ V n : g Þ Ñ g¨v is a graph isomorphism between the restriction of CaypG, Sq to C and the restriction of Γ n to C¨v. Then for any α P Z 1 pΓ n , Tq and any v P V n we have pγps, g¨vqq sPS,gPE : γ P α`B 1 pΓ n , Tq Proof. We first reduce to the case α " 0. To this end, observe that if α : E n ÝÑ T is a cocycle, then it has zero sum around any based loop in Γ n which corresponds to a relation of the group. The map defines an isomorphism of the restricted graphs by assumption, so any loop of Γ n that is contained in C¨v must arise from a relation of the group. Therefore the cocycle condition implies that α sums to zero around any loop contained in C¨v, and hence pα| Sˆg¨v q gPE " ppdβq| Sˆg¨v q gPE for some β P T C¨v obtained by simply summing along paths from some distinguished basepoint in C: all of C can be reached this way because C is connected.
Extending β arbitrarily to a member of T Vn , this shows that the left-hand side of (20) does not depend on α: for every α that left-hand side is a coset of a homomorphic image of B 1 pΓ n , Tq, and we have just seen that any two of these cosets overlap, hence are equal. Now suppose that α " 0. We will prove two separate inclusions. First, if θ P T G and γ " π E pΦpθqq " pθ sg´θg q sPS,gPE P π E pZq, then γ is also equal to ppdβq| Sˆg¨v q gPE for any choice of β P T Vn satisfying β g¨v " θ g @g P E.
Such a choice exists because C Ě E and the map (21) is injective.
On the other hand, if β P T Vn , then the reverse of this argument produces some θ P T G such that β g¨v " θ g for all g P E. Proof. For α P Z 1 pΓ n , Tq, let R α : T Vn ÝÑ T Vn : β Þ Ñ β`α be the corresponding rotation. We need to show that the image measures pR αn q˚pϕ σn q˚mˆV n are asymptotically supported on ΩpO, σ n q as n ÝÑ 8 for any sequence α n P Z 1 pΓ n , Tq. We will deduce this by showing that pR αn q˚pϕ σn q˚mˆV n q ÝÑ µ Z , where this refers to quenched convergence as in [1,Definition 5.3].
Since pT SˆG , µ Z , Sq is a factor of a Bernoulli shift, it is ergodic. Therefore by [1,Corollary 5.7] quenched convergence will follow if we show that pR αn q˚pϕ σn q˚mˆV n lwÝ Ñ µ Z (local weak˚convergence).
The measure pR αn q˚pϕ σn q˚mˆV n is equal to the Haar measure on the coset B 1 pΓ n , Tq`α n .
The result now follows from Lemma 6.6. According to that lemma, for any finite E Ď G, the projection of this coset to the directed edges which emanate from E¨v is simply a copy of π E pZq, provided n is large enough depending on E. For such n, the projection of pR αn q˚pϕ σn q˚mˆV n to E¨v is therefore equal to µ π E pZq " pµ Z q E .
Proof of Theorem D. Let r be as in Theorem 6.1, let δ :" r{4, and let ε ą 0 and F Ď R be given by part 2 of Corollary 6.3 for r 1 :" r{10. Finally, let O be a w˚-neighbourhood of µ Z as given by Lemma 6.5 for this F and ε.
For any w˚-neighbourhood U of µ Z , Lemma 6.7 gives that ΩpU, σ n q X pα`B 1 pΓ n , Tqq ‰ H for any α P Z 1 pΓ n , Tq, once n is sufficiently large. Next, for each n, we may identify G ñ T G{Gn as the induction to G of the trivial action of G n on T. Hence Shapiro's Lemma [15,Theorem II.3.7] gives H 1 pG, T G{Gn q -H 1 pG n , Tq " HompG n , Tq "`G n {rG n , G n s˘^‰ 0, where the last conclusion is the point at which we use part (i) of Lemma 6.4. Therefore for each n there is more than one coset of B 1 pΓ n , Tq in Z 1 pΓ n , Tq.
Since α n,k P Z 1 ε,F pΓ n , Tq, this would contradict part 2 of Corollary 6.3.

Complemented factors
The deduction of Corollary D 2 rests on the following theorem about a general metric G-process pX G , µ, S, d X q, which may be of independent interest. It is expressed in terms of a sequence of measures on the spaces X Vn which 'doublyquenched converge' to µ P Prob S pX G q over Σ, denoted by µ n dq ÝÑ µ. This property is introduced in [1, Subsection 5.2]. By [1,Corollary 5.18], the existence of such a sequence µ n is an isomorphism-invariant of the process which does not depend on the metric d X . We do not recall the definition here, but refer the reader to that reference for full details. Theorem 6.8. Suppose there exist µ n P ProbpX Vn q such that µ n dq ÝÑ µ rel Σ. Then any complemented factor map of pX G , µ, Sq is model-surjective rel Σ.
In particular, any complemented factor map of a Bernoulli shift is modelsurjective relative to any sofic approximation of G.
Proof. Suppose that Φ : pX G , µ, S, d X q ÝÑ pY G , ν, S, d Y q is a complemented factor map, so there is another factor map Ψ : pX G , µ, S, d X q ÝÑ pZ G , θ, S, d Z q such that the combined map pΦ, Ψq : pX G , µ, Sq ÝÑ pY GˆZ G , νˆθ, Sq is a measure-theoretic isomorphism.
By [1,Proposition 5.16], there are also measures θ n P ProbpZ Vn q such that θ n dq ÝÑ θ rel Σ: they can be obtained as images of the measures µ n . Using these, [1,Corollary 5.13] gives that the coordinate-projection factor map Π : pY GˆZ G , νˆθ, Sq ÝÑ pY G , ν, Sq is model-surjective rel Σ. This conclusion is written in terms of w˚-neighbourhoods in [1], but it is easily turned into condition 2 of Lemma 3.4. Therefore the factor map Φ " Π˝pΦ, Ψq is a composition of a factor map which is model-surjective rel Σ and an isomorphism, so Φ itself is also model-surjective rel Σ by Propositions 3.5 and 3.6. This argument applies whenever pX G , µ, Sq is a Bernoulli shift and Σ is a sofic approximation, because then the corresponding product measures on the spaces X Vn always doubly-quenched converge to µ [1, Lemma 5.11].
Proof of Corollary D 2 . This follows at once from the combination of Theorems A, C, D and 6.8.

Remark.
It is easy to prove that the Popa factor map itself is not complemented for any infinite discrete group G. This is because it is a relatively compact extension, so if it had a complementing factor Ξ : pT G , mˆG, Sq ÝÑ pZ, θ, Rq then pZ, θ, Rq would have to be a compact system, but the original Bernoulli shift pT G , mˆG, Sq has no compact factors.
Corollary D 2 is much stronger: it prohibits the Popa factor system from appearing as a complemented factor of any other Bernoulli system in any way. This holds only for certain special groups G. By contrast, for amenable G the Popa factor is always isomorphic to another copy of the Bernoulli shift by the general theory of [17], and the same holds if G is a free group by an easy calculation. ⊳