Approximate homomorphisms and sofic approximations of orbit equivalence relations

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphism. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.


Introduction
This paper is a follow-up to [32].We are once again interested in approximate homomorphisms of groups.Recall that approximate homomorphisms are sequences of maps σ n : G → H n where G, H n are groups, each H n has a bi-invariant metric d n and the sequence satisfies lim n→∞ d n (σ n (gh), σ n (g)σ n (h)) = 0 for all g, h ∈ G.
In our previous article, we studied automorphic conjugacy of sofic approximations.In this article, we are interested in when approximation homomorphisms with values in permutation groups are asymptotically conjugate via a sequence of permutations.
One motivation for the study of asymptotic conjugacy of approximate homomorphisms of groups is the notion of stability.Stability of homomorphisms dates back to the 40's via work of Hyers [33] (answering a question of Ulam), and asks when approximate homomorphisms can be perturbed to a sequence of honest homomorphisms.E.g. we say a group is permutation stable if every sequence of approximate homomorphisms of the group with values in permutation groups is pointwise close to a sequence of honest homomorphisms.See [4,5,7,23,24,30,34], for work in this direction.The notion of stability gives a potential approach to proving the existence of a nonsofic group [29,4,11], has connections to cohomology [21,20,19], and to operator algebras [13,18,31,6].
While not immediate, there is a natural connection between stability and asymptotic conjugacy.For example, one could consider a modification of permutation stability where we only demand that every sofic approximation (instead of every approximate homomorphism) is close to an honest homomorphism (this concept is introduced in [4] under the name weak stability).When the group is amenable, then as shown in [4, Theorem 1.1], the Kerr-Li [36,Lemma 4.5] and Elek-Szabo [27] uniqueness theorems imply that this modified version of permutation stability is equivalent to residual finiteness.
The situation for permutation stability of general asymptotic homomorphisms of amenable groups was fully classified by Becker-Lubotzky-Thom [7,Theorem 1.3] who showed that an amenable group G is permutation stable if and only if every invariant random subgroup of G is a limit of IRS's coming from actions on finite sets.Invariant random subgroups arose from the works [2,9,47,49], we recall the definition in 2.1.A different proof of this was given in [5,Theorem 3.12], which reformulates the results in terms of traces.
Given a sequence (σ n ) n of approximate homomorphisms and a free ultrafilter ω, one can naturally produce an invariant random subgroup associated to (σ n ) n , ω which we denote by IRS(σ ω ) (see Definition 2.4).Given an invariant random subgroup Θ of G, following [2, Proposition 13], we can construct a generalized Bernoulli shift action G (X Θ , µ Θ ) associated to Θ. Let R G,X Θ be the orbit equivalence relation of this action.We show that for every sequence of asymptotic homomorphisms of G whose given IRS is Θ, we may extend the asymptotic homomorphism to a sofic approximation (in the sense of [26]) of R G,X Θ .This extension result holds for any countable group G.
Theorem 1.1.Let G be a group, and let (σ n ) ∞ n=1 be a sequence of approximate homomorphisms of G and ω a free ultrafilter on N. Let σ ω be the ultraproduct of this sequence.Let Θ be the IRS of (σ ω ), and let R G,X Θ be the orbit equivalence relation of the Θ-Bernoulli shift over G with base [0, 1].Then σ ω extends to a sofic approximation of R G,X Θ .
In the case where Θ is δ {1} , our result says that any sofic approximation of G extends to a sofic approximation of any Bernoulli shift over G.This was previously proved in [26,42,44,10] (in fact, [44] proves that the generalized Bernoulli shift R G (X,µ) G/H is sofic if G is sofic, H is amenable and (X, µ) is any probability space).The proof of the case Θ = δ {1} given in [26,Proposition 7.1] (see also [14,Theorem 3.1] and [10,Theorem 8.1]) provided inspiration for the proof we give of Theorem 1.1.
In the context of Theorem 1.1, if G is amenable, then R G,X Θ is hyperfinite by Ornstein-Weiss [40], [41, II §3] (see Connes-Feldman-Weiss [15] for more general results), and thus any two sofic approximations of R G,X Θ are conjugate [42,Proposition 1.20].Using this conjugacy fact, the results of [5, Theorem 3.12], [7,Theorem 1.3] are corollaries of Theorem 1.1.This again illustrates the utility of approximate conjugacy results in the context of stability.
Comments on some applications.To demonstrate the utility of Theorem 1.1 we recover with separate proof ideas, some results characterizing conjugacy for almost representations of amenable group appearing in [39,25,4,7].Namely, those results are proved via usage of the asymptotic combinatorial notation of hyperfinite graphs, and on Benjamini-Schramm convergence.Our proof is in some sense more "continuous", and is based purely in ergodic theory and probabilistic arguments.In particular, we do not need the combinatorial notion of hyperfiniteness of graph sequences or Benjamini-Schramm convergence.In some sense, one can view ergodic theory as a limit of combinatorics and so our methods can be viewed as a limiting version of those in [39,25,5,7].Both the combinatorial and continuous approaches have their utility.We believe one benefit of our approach is that it reveals that one can work directly with the limiting object, and does not always have to resort to working with combinatorics at the finitary level and taking limits.
Organization of the paper.We begin in Section 2 by recalling some background on metric groups and approximate homomorphisms.We also give the definition of the IRS of a sequence of approximate homomorphisms here and restate Theorem 1.1 in these terms.In Section 3, we recall the background on orbit equivalence relations and their sofic approximations we need and state Theorem 1.1 in these terms.In Section 5 we deduce [5, Theorem 3.12] from 1.1, and also deduce [7,Theorem 1.3] from Theorem 1.1.We also explain the connection between action traces and IRS's in this section.In Section 4, we prove Theorem 1.1.

Preliminaries
Throughout we consider G to be a countable group.Let Sym(n) denote the finite symmetric group of rank n.The normalized Hamming distance, which is a bi-invariant metric on Sym(n), is given by Recall the following.
Definition 2.1.A sequence of maps σ n : G → Sym(d n ) are said to be approximate homomorphisms if for all g, h ∈ G we have lim Let ω be a free ultrafilter on N. Let (G n , d n ) be countable groups with bounded bi-invariant metrics.Denote by Observe that by the bi-invariance property of the metrics its image in the ultraproduct.In the above context, a sequence of approximate homomorphisms For a nonnegative integer k, we use Suppose that X is a compact, metrizable space.Assume (Y, ν) is a standard probability space and that we have a Borel map Y → Prob(X) given by y → µ y .Then by the Riesz representation theorem, we can define Y µ y dν(y) to be the unique probability measure η satisfying for all f ∈ C(X).

2.1.
Preliminaries on IRS's.Given a sequence of approximate homomorphisms σ n : G → Sym(d n ), define S σn : [d n ] → {0, 1} G by (S σn )(j)(g) = 1 {j} (σ n (g)(j)).Throughout the paper, for a finite set E, we use u E for the uniform measure on E and we typically use u d instead of u [d] .Set Θ n = (S σn ) * (u dn ).It turns out that subsequential limits of Θ n can be nicely described in terms of well known objects.Definition 2.2.Let G be a countable, discrete group.We let Sub(G) be the set of subgroups of G, which we regard as a subspace of {0, 1} G by identifying a subgroup with its indicator function.We equip Sub(G) with the topology induced from this inclusion.Then Sub(G) is a closed subset of {0, 1} G and is thus compact in the induced topology.We let IRS(G) be the set of probability measures on Sub(G) which are invariant under the conjugation action of G given by g • H = gHg −1 for all g ∈ G, H ∈ Sub(G).
We equip Prob({0, 1} G ) with the weak * -topology (viewing complex Borel measures on {0, 1} G as the dual of C({0, 1} G )).We often regard Prob(Sub(G)) as a closed subset of Prob({0, 1} G ) by identifying Prob(Sub(G)) with the probability measures which assign Sub(G) total mass one.In [2, Proposition 13], it is shown that Θ ∈ IRS(G) if and only if there is a probability measure-preserving action G (X, µ) so that Θ = Stab * (µ) where Stab : X → Sub(G) is given by Stab(x) = {g ∈ G : gx = x}.So IRS's are a naturally occurring construction when considering general (i.e.not assumed essentially free) probability measure-preserving actions.One can think of a sofic approximation as a sequence of almost free almost actions on finite sets.From this perspective, it is reasonable to expect IRS's to arise when one considers more general almost actions (i.e.asymptotic homomorphisms) that are not asymptotically free almost actions.The following lemma explains exactly how IRS's arise from approximate homomorphisms.Lemma 2.3.Let G be a countable, discrete group and σ n : G → Sym(d n ) approximate homomorphisms.For every free ultrafilter ω ∈ βN \ N we have that Proof.Let G {0, 1} G be given by (g • x)(h) = x(g −1 hg).We use α for the induced action on functions: Then if g, h ∈ F and j ∈ Ω n , we have: This shows that: (2.1) ) and permutation invariance of u dn that: ) and so by the Riesz representation theorem we have shown that Θ ω is invariant under the conjugation action of G.
It thus suffices to show that Θ ω is supported on the space of subgroups of G.For g, h ∈ G let: Since G is countable, it suffices to show that Θ ω assigns each set in this intersection measure 1.By the Portmanteau theorem [22,Theorem 11.1.1],for each g, h ∈ G: The proofs that Θ ω assigns measure 1 to I g and {x ∈ {0, 1} G : x(e) = 1} are similar.
We are thus able to make the following definition.We remark on an alternate construction of IRS(σ ω ).One can take an ultraproduct of the measure spaces ({1, • • • , d n }, u dn ) to obtain a probability space (L, u ω ) called the Loeb measure space [38].The actions Sym(d n ) {1, • • • , d n } along with the approximate homomorphisms σ n will induce a probability measure-preserving action on (L, u ω ) in a natural way.Under this action, one can show that IRS(σ ω ) is Stab * (u ω ) where Stab : L → Sub(G) is given by Stab(z) = {g ∈ G : gz = z}.However, we will not need this fact and thus will not prove it.If IRS(σ ω ) does not depend upon ω, then lim n→∞ Θ n exists.In this case, we call lim n→∞ Θ n the stabilizer type of σ n .E.g. lim n→∞ Θ n = δ {1} if and only if σ n is a sofic approximation.

Background on Orbit equivalence relations and Theorem 1.1
In order to extend approximate homomorphisms to sofic approximations of relations, we need the following construction (appearing first in [2]) of an action associated to an IRS, say Θ.The intention is that the action is "Bernoulli as possible" while still having Θ as its IRS.For technical reasons the action will not always have Θ as its IRS, but under mild conditions (which we state precisely after the definition) it will, and we think it is still worth stating the general construction.Definition 3.1.Let G be a countable, discrete group and let Θ ∈ IRS(G).Let X be a compact metrizable space and ν a Borel probability measure on G. Let H be the space of right cosets of H in G and regard ν ⊗G/H as a probability measure on X G which is supported on the x ∈ X G which are constant on right H-cosets.Let µ Θ be the measure We sometimes denote (Y, µ Θ ) as Bern(X, ν, Θ).We let G X G by Suppose that ν is not a dirac mass, and choose b Since b < 1, and the above inequality is true for every finite F ⊆ G, we deduce that if either So, under either of the above bulleted conditions, the action G (Y, µ Θ ) has IRS equal to Θ.This construction first appears in [1, Proposition 13] (see [17, for the locally compact case, as well as a proof of the fact that if Θ is ergodic then we can modify the above construction to get an ergodic action).See [46], [48,Section 5] for further applications of this construction.
The classical Bernoulli shift with base (X, ν) is the case when Θ = δ {1} .It is of great importance in ergodic theory, via its connections to probability (it is the sample space for i.i.d.X-valued random variables (Υ g ) g∈G ).It also has many desirable properties such as being mixing ([43, Section 2.5] and [37, Section 2.3]), complete positive entropy [45,35], Koopman representation being an infinite direct sum of the left regular [37, Section 2.3], and being a free action when the acting group is infinite (proved above).It is also canonically associated to any group.We refer the reader to [50, Section 4.9], [43, Section 6.4 and 6.5], and [37, Section 2.3] for more details and information on the classical Bernoulli shift.For our purposes, we will only need that the Θ-Bernoulli shift retains a residue of freeness, in that under the above conditions it has Θ as its IRS.
We will need the notion of an orbit equivalence relation.
where (X, ν) is a standard probability space, R ⊆ X × X is Borel, and so that the following holds: • (equivalence relation) the relation x ∼ y given by (x, y) ∈ R is an equivalence relation, • (discreteness) for almost every x ∈ X, we have The last item can be recast as follows: define a Borel measure ν on R by Then the map (x, y) → (y, x) preserves the measure if and only if the last item holds.This implies, for example, that if R is a discrete, probability measure preserving equivalence relation on (X, ν) If G is a countable discrete group and G (X, ν) is a probability measure preserving action, we then have a discrete, probability-measure relation given as the orbit equivalence relation All discrete, probability measure-preserving equivalence relations arise this way [28].
As mentioned before, we will extend approximate homomorphisms to sofic approximations of equivalence relations.In order to define a sofic approximation of an equivalence relation, we use tracial von Neumann algebras.Definition 3.3.Let H be a Hilbert space.A unital * -subalgebra M of B(H) is said to be a von Neumann algebra if it is closed in the weak operator topology given by the convergence We let Proj(M ) be the set of projections in M .A normal homomorphism between von Neumann algebras M, N is a linear π : M → N which preserves products and adjoints and such that π {x∈M : x ≤1} is weak operator topology continuous.Such maps are automatically norm continuous [16, Proposition 1.7 (e)].We say that π is an isomorphism if is bijective, it is then automatic that π −1 is a normal homomorphism [16,Proposition 46.6].A pair (M, τ ) is a tracial von Neumann algebra if M is a von Neumann algebra and τ is a trace, meaning that τ : M → C satisfies: Given a Hilbert space H and E ⊆ B(H), we let W * (E) be the von Neumann algebra generated by E.
For a tracial von Neumann algebra (M, τ ) and x ∈ M , we set x 2 = τ (x * x) 1/2 .A simple example of a tracial von Neumann algebra is the following.For k ∈ N, define tr : Then (M k (C), tr) is a tracial von Neumann algebra.The following is folklore, but we highlight it because we will use it explicitly.This can be proved, e.g. by following the discussion in Section 2 of [8].
Lemma 3.4.Let (M j , τ j ), j = 1, 2 be tracial von Neumann algebras.Suppose that N ⊆ M 1 , is a weak operator topology dense * -subalgebra, and that π : Given a discrete, probability measure-preserving equivalence relation R on (X, ν), we let [R] be the group of all bimeasurable bijections γ : X 0 → Y 0 where X 0 , Y 0 are conull subsets of X and with γ(x) ∈ [x] R for almost every x ∈ X.As usual, we identify two such maps if they agree off a set of measure zero.The group [R] is called the full group of R. We let [[R]] be the set of all bimeasurable bijections γ : B 1 → B 2 (where B 1 , B 2 are measurable subsets of X) which satisfy that γ(x) ∈ [x] R for almost every x ∈ B 1 .As usual, we identify two such maps if they agree off a set of measure zero.We usually use dom(γ), ran(γ) for B 1 , B 2 above.We define maps ϑ : We define the von Neumann algebra of the equivalence relation to be The von Neumann algebra L(R) is equipped with a trace where Λ = {(x, x) : x ∈ X}.We typically identify L ∞ (X, ν) and [[R]] as subsets of L(R) and do not make explicit reference to the maps λ, ϑ.
Another example of a tracial von Neumann algebra is the ultraproduct of tracial von Neumann algebras.Let ω be a free ultrafilter on N. Suppose (N k , τ k )are tracial von Neumann algebras.Denote the ultraproduct by By the proof of [12, Lemma A.9] the ultraproduct is a tracial von Neumann algebra and is equipped with a canonical trace τ (( For a sequence of integers d n , set We let and View ℓ ∞ (d n ) ⊆ M dn (C) by identifying each function with the diagonal matrix whose entries are , and identify each permutation with its corresponding permutation matrix.In this way we can identify L ∞ (L, u ω ) as a subalgebra of M and S ω as a subgroup of the unitary group of M.
Definition 3.5.Let (X, ν, R) be a discrete, probability measure preserving equivalence relation.We say that R is sofic if there is a free ultrafilter ω, a sequence of positive integers (d n ) n and maps so that: Since [R], L ∞ (X, ν) are uncountable, this definition can be a bit unwieldy.We give a few equivalent definitions of soficity below for the readers convenience.Essentially all of this is either folklore or due to Elek-Lippner (see the proof [26, Theorem 2]) or Pȃunescu ([42]), we do not claim originality for these results.Proposition 3.6.Let R be a discrete, probability measure-preserving relation over a standard probability space (X, ν).View L ∞ (X, ν) and [[R]] as subsets of L(R).
(ii) Conversely, suppose that π : (iii) Let D ⊆ L ∞ (X, ν) be a subset which is closed under products, and Then there is a unique sofic approximation (ρ, σ) of R so that ρ| D = ρ 0 , σ| G = σ 0 .(iv) Let A be an algebra of measurable sets in X, and let G be a countable subgroup of [R] with Gx = [x] R for almost every x ∈ X. Assume that A is G-invariant and that the complete sigma-algebra generated by A is the algebra of all ν-measurable sets.Suppose that ρ 0 : A → Proj(L ∞ (L, u ω )) and σ 0 : G → S ω satisfies: Then there is a unique sofic approximation ρ : then M 0 is weak operator topology dense in L(R) by definition.Moreover, for all we have by the axioms of a sofic approximation: .
Since • 2 is a norm, the above calculation implies that j ρ(f j )σ(γ j ) = 0 if and only if j f j λ(γ j) = 0.This implies that the map π 0 : M 0 → k→ω (M k (C), tr) given by is a well-defined linear map.It is direct to check from the definition of a sofic approximation that it is a trace-preserving * -homomorphism.Lemma 3.4 implies that π 0 has a unique extension to trace-preserving normal * -homomorphism π.
(ii): This is an exercise in understanding the definitions.(iii): Let A = span(D) so that A is a weak operator topology dense * -subalgebra of L ∞ (X, ν).Let As in (i) we know there is a unique function π 0 : M 0 → M satisfying Our hypothesis imply that π 0 is a trace-preserving * -homomorphism.Thus by Lemma 3.4 and items (i),(ii) it suffices to show that M 0 is weak * -dense in L(R).Let M be the weak operator topology closure of M 0 .Then ] we may find (not necessarily unique) disjoint sets (B g ) g∈G so that B g ⊆ {x ∈ dom(γ) : γ(x) = gx} and with We leave it as an exercise to check that with the sum converging in the strong operator topology.Since we have already shown that Because Proposition 3.6 gives several equivalent definitions of soficity, we will often use the term sofic approximation to any one of the kinds of maps in each item of this proposition.For example, a map π : L(R) → k→ω (M k (C), τ ) satisfying the hypotheses of (i) will be called a sofic approximation.
Additionally, item (iv) suggest a sequential version of a sofic approximation of an equivalence relation.Namely, we can consider sequences ρ n : A → P({1, • • • , d n }) and σ n : G → Sym(d n ) so that: • σ n is an asymptotic homomorphism, This is sometimes taken as the definition of soficity.Our definition has the advantage of being canonical and not requiring a choice of G, A. However, this alternate definition is typically how one would check soficity of relations in specific examples, whereas ours is more abstract.Similarly, item (iii), suggests a different definition of soficity.One could require a sequence of maps Here the • 2 -norms are with respect to the uniform measure and α g (f )(x) = f (g −1 x).We will refer to either of these as a sofic approximation sequence.
We now rephrase Theorem 1.1.We use m for the Lebesgue measure on [0, 1].
4. The proof of Theorem 3.7 We now proceed to prove Theorem 3.7.We remark that one can use [48,Theorem 1.5] to prove Theorem 3.7.The essential idea behind such a proof is that given a sequence of approximate homomorphisms with IRS Θ, the induced action on the Loeb measure space G (L, u ω ) also has IRS Θ.One can use the Löwenheim-Skolem theorem in continuous model theory to build an action on a standard probability space which is a factor of this action and which still has IRS Θ. Being an action on a standard probability space, such a factor is weakly contained in a Θ-Bernoulli shift by [48,Theorem 1.5].This implies that the Θ-Bernoulli shift action G (X Θ , µ Θ ) with base [0, 1] (equipped with Lebesgue measure) is weakly contained in G (L, u ω ).Since G (L, u ω ) is an ultraproduct action, if G (X Θ , µ Θ ) is weakly contained in G (L, u ω ) it must actually be a factor of this action.Putting this altogether shows that any sequences of approximate homomorphisms extends to a sofic approximation of the Θ-Bernoulli shift.
For the sake of concreteness, we have instead elected to give a direct probabilistic argument for the proof of Theorem 3.7.The following is the main technical probabilistic lemma we need and is inspired by the proof of [10,Theorem 8.1].Bowen's purpose in [10] for such a result was to prove that the sofic entropy of Bernoulli shifts is the Shannon entropy of the base space.In this regard our proof should be compared with [46,Theorem 9.1] which also says, in some sense, that the Θ-Bernoulli shift with base (X, ν) is the "largest entropy" action whose IRS is Θ and which is generated by the translates of an X-valued random variable with distribution ν.
For ease of notation, if X is a compact, metrizable space, ν ∈ Prob(X), and f : X → C is bounded and Borel we often use ν(f ) for f dν.Suppose I is a set, and (X i ) i∈I are compact Hausdorff spaces.If E ⊆ I is finite, and compact, metrizable space we often slightly abuse notation and regard Then: Proof.Throughout, we use G/H for the space of right cosets of H in G.For F ⊆ G finite, we set By Stone-Weierstrass, span(̥) is norm dense in C(({0, 1}×[0, 1]) G ), and by the Riesz representation theorem to show that η = µ Θ it suffices to show that they have the same integral against any element of ̥.
Since σ n is an approximate homomorphism u dn (Ω n ) → 1. Hence: Since u dn (Ω n ) → 1, and the f g are bounded, the first term tends to zero.It thus remains to analyze the second term.For j ∈ Ω n , , where in the last step we use that f g ≥ 0 to make sense of the fractional power (which is there to account for over-counting).Because j ∈ Ω n : is continuous (in fact, it is locally constant).The definition of IRS(σ ω ) thus proves that:

By definition,
where we view m ⊗G/H as a probability measure on [0, 1] G which is supported on the x ∈ [0, 1] G which are constant on right H-cosets.So Altogether, this shows that Giving C(({0, 1} × [0, 1]) G ) the supremum norm we have the operator norm of T n satisfies T n ≤ 2. This uniform estimate implies that {ζ ∈ C(({0, 1} × [0, 1]) G ) : T n ζ 2 → n→ω 0} is a closed, linear subspace.So as in (i), it suffices to verify the desired statement for f = g∈E f g E F ∈ B, where (f g ) g∈E ∈ C([0, 1]) G and E, F ⊆ G are finite.By direct computation, (4.1) We can rewrite this as For a fixed j, the set of k for which σ n (E) −1 (j) ∩ σ n (E) −1 (k) = ∅ has cardinality at most |E| 2 .So the first term is bounded by The second term in (4.2) is: By another direct computation, By the same estimates as above, Combining this with (4.1), (4.3) shows that Since this is true for every free ultrafilter, we have proven (ii).
(iii): Since f is continuous, we can find an ) G and continuity of f imply that we may find a finite E ⊆ G and a δ > 0 so that if x, y ∈ ({0, 1} × [0, 1]) G and x(g) − y(g) < δ for all g ∈ E, Then, for any x ∈ [0, 1] dn we have Since σ n is an asymptotic homomorphism the second term tends to zero as n → ∞.
While technical, Lemma 4.1 has all the tools to prove Theorem 3.7.Indeed, as we now show, Lemma 4.1 essentially says that a random choice of x ∈ [0, 1] dn will produce a sofic approximation of the appropriate equivalence relation.
Proof of Theorem 3.7.For notation, define S : . Hence if k ≥ (2023)!and n ∈ L k , we may choose an x n ∈ Ω n .Let x n be defined arbitrarily for n ∈ N \ L (2023)! .Define Then ρ 0 preserves products.Recall that we view ℓ ∞ (d n ) as a subalgebra of M dn (C) by identifying each function with the corresponding diagonal matrix.Let f ∈ D, g ∈ G.
By our choice of x n , where in the last step we use (4.4).By Lemma 4.1 this last limit is Since ([0, 1], m) is atomless, and G is countable Thus for every g ∈ G: Thus Proposition 3.6 (iii) implies that implies there is a unique sofic approximation π : . By Proposition 3.6 (ii) we have that (ρ, σ) is a sofic approximation.The fact that σ •Ξ = σ ω is true by construction.

Applications of Theorem 1.1
One of the applications of the Theorem we wish to highlight is the following result which is a reformulation of the Newman-Sohler Theorem [39] (see [25,Theorem 5] for a statement of the Newman-Sohler theorem which is closer to our language).Theorem 5.1.Let G be an amenable group, and σ n , ψ n : G → Sym(d n ) two sequences of approximate homomorphisms.Let ω ∈ βN \ N. Then σ ω is conjugate to ψ ω if and only if IRS(σ ω ) = IRS(ψ ω ).
In particular, is in ω and by our choice of n k , ω it follows that L ∈ ω c .This contradicts ω being a filter.
For The proof of the reverse implication given above is similar to the one in [5,Proposition 3.15].

,
with the last step following because the inverse image of g under the map E → G/H, g → Hg has cardinality |{a ∈ E : Ha = Hg}|.Thus a decreasing sequence of sets and Lemma 4.1 implies that L k ∈ ω for all k ∈ N, and k L k = ∅ by the first bullet point.Set L 0