2. Preliminaries

Throughout, we consider G to be a countable group. Let ${\operatorname {Sym}}(n)$ denote the finite symmetric group of rank n. The normalized Hamming distance, which is a bi-invariant metric on ${\operatorname {Sym}}(n)$ , is given by

$$ \begin{align*} d_{\operatorname{\mathrm{Hamm}}}(\chi_{1}, \chi_{2})= \frac{|\{i: \chi_{1}(i)\neq \chi_{2}(i)\}|}{n}. \end{align*} $$

Recall the following.

Definition 2.1. A sequence of maps $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ are said to be approximate homomorphisms if for all $g,h\in G$ , we have

$$ \begin{align*}\lim_{n\to \infty} d_{\operatorname{\mathrm{Hamm}}}(\sigma_n(gh), \sigma_n(g)\sigma_n(h))=0.\end{align*} $$

Let $\omega $ be a free ultrafilter on $\mathbb {N}$ . Let $(G_n,d_n)$ be countable groups with bounded bi-invariant metrics. Denote by

$$ \begin{align*} \prod_{n\to \omega} (G_n,d_n)= \{(g_n)_{n\in \mathbb{N}}\}/\{(g_n): \lim_{n\to \omega} d_n(g_n,1_{G_n})=0\}. \end{align*} $$

Observe that by the bi-invariance property of the metrics $d_n$ , the subgroup $\{(g_n): \lim _{n\to \omega } d_n(g_n,1_n)=0\}$ is a normal subgroup. If $(g_{n})_{n}\in \prod _{n}G_{n},$ we let $(g_{n})_{n\to \omega }$ denote its image in the ultraproduct. In the above context, a sequence of approximate homomorphisms $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ naturally produces a homomorphism into $\prod _{n\to \omega }({\operatorname {Sym}}(d_{n}),d_{\operatorname {\mathrm {Hamm}}})$ by

$$ \begin{align*} \sigma_{\omega}(g)=(\sigma_{n}(g))_{n\to\omega}. \end{align*} $$

For a non-negative integer k, we use $[k]=\{1,\ldots ,k\}$ . We also set $[0]=\varnothing $ .

Suppose that X is a compact, metrizable space. Assume $(Y,\nu )$ is a standard probability space and that we have a Borel map $Y\to \operatorname {\mathrm {Prob}}(X)$ given by $y\mapsto \mu _{y}$ . Then by the Riesz representation theorem, we can define $\int _{Y}\mu _{y}\,d\nu (y)$ to be the unique probability measure $\eta $ satisfying

$$ \begin{align*} \int_{X} f\,d\eta=\int_{Y}\int_{X}f\,d\mu_{y}\,d\nu(y) \end{align*} $$

for all $f\in C(X)$ .

2.1. Preliminaries on IRSs

Given a sequence of approximate homomorphisms $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ , define $S_{\sigma _{n}}\colon [d_{n}]\to \{0,1\}^{G}$ by $(S_{\sigma _{n}})(j)(g)=1_{\{j\}}(\sigma _{n}(g)(j))$ . 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 $\Theta _{n}=(S_{\sigma _{n}})_{*}(u_{d_{n}})$ . It turns out that subsequential limits of $\Theta _{n}$ can be nicely described in terms of well-known objects.

Definition 2.2. Let G be a countable, discrete group. We let $\operatorname {\mathrm {Sub}}(G)$ 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 $\operatorname {\mathrm {Sub}}(G)$ with the topology induced from this inclusion. Then $\operatorname {\mathrm {Sub}}(G)$ is a closed subset of $\{0,1\}^{G}$ and is thus compact in the induced topology. We let $\operatorname {\mathrm {IRS}}(G)$ be the set of probability measures on $\operatorname {\mathrm {Sub}}(G)$ which are invariant under the conjugation action of G given by $g\cdot H=gHg^{-1}$ for all $g\in G, H\in \operatorname {\mathrm {Sub}}(G)$ .

We equip $\operatorname {\mathrm {Prob}}(\{0,1\}^{G})$ with the weak $^{*}$ -topology (viewing complex Borel measures on $\{0,1\}^{G}$ as the dual of $C(\{0,1\}^{G})$ ). We often regard $\operatorname {\mathrm {Prob}}(\operatorname {\mathrm {Sub}}(G))$ as a closed subset of $\operatorname {\mathrm {Prob}}(\{0,1\}^{G})$ by identifying $\operatorname {\mathrm {Prob}}(\operatorname {\mathrm {Sub}}(G))$ with the probability measures which assign $\operatorname {\mathrm {Sub}}(G)$ total mass one. In [Reference Abért, Glasner and Virág2, Proposition 13], it is shown that $\Theta \in \operatorname {\mathrm {IRS}}(G)$ if and only if there is a probability measure-preserving action $G\,{\curvearrowright }\, (X,\mu )$ so that $\Theta ={\operatorname {Stab}}_{*}(\mu )$ , where ${\operatorname {Stab}}\colon X\to \operatorname {\mathrm {Sub}}(G)$ is given by ${\operatorname {Stab}}(x)=\{g\in G:gx=x\}$ . So IRSs are a naturally occurring construction when considering general (that is, 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 IRSs to arise when one considers more general almost actions (that is, asymptotic homomorphisms) that are not asymptotically free almost actions. The following lemma explains exactly how IRSs arise from approximate homomorphisms.

Lemma 2.3. Let G be a countable, discrete group and $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ approximate homomorphisms. For every free ultrafilter $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ , we have that

$$ \begin{align*}\Theta_{\omega}=\lim_{n\to\omega}\Theta_{n}\in \operatorname{\mathrm{IRS}}(G).\end{align*} $$

Proof. Let $G\,{\curvearrowright }\, \{0,1\}^{G}$ be given by $(g\cdot x)(h)=x(g^{-1}hg)$ . We use $\alpha $ for the induced action on functions: $(\alpha _{g}f)(x)=f(g^{-1}\cdot x)$ for all $f\colon \{0,1\}^{G}\to {\mathbb C}$ Borel and all $g\in G$ . Note that if $F\subset G$ is finite, then we can find a sequence $\Omega _{n}\subseteq [d_{n}]$ (depending upon F) with $u_{d_{n}}(\Omega _{n})\to 1$ so that for all $j\in \Omega _{n}$ , all $1\leq k\leq (2023)!$ , all $g_{1},\ldots ,g_{k}\in F\cup F^{-1}\cup \{e\}$ , and all $s_{1},\ldots ,s_{k}\in \{\pm 1\}$ , we have

$$ \begin{align*} \sigma_{n}(g_{1}^{s_{1}}\cdots g_{k}^{s_{k}})(j)=\sigma_{n}(g_{1})^{s_{1}}\cdots\sigma_{n}(g_{k})^{s_{k}}(j). \end{align*} $$

Then if $g,h\in F$ and $j\in \Omega _{n}$ , we have

$$ \begin{align*}(g\cdot S_{n}(j))(h) =S_{n}(\sigma_{n}(g)j)(h).\end{align*} $$

This shows that

(2.1) $$ \begin{align} \lim_{n\to\infty}u_{d_{n}}\bigg(\bigcap_{h\in E}\{j:(g\cdot S_{n}(j))(h)=S_{n}(\sigma_{n}(g)j)(h)\}\bigg)=1 \quad \text{for all } g\in G,E\subseteq G \text{ finite}. \end{align} $$

For a finite $E\subseteq G$ , let $A_{E}=\{f\circ \pi _{E}:f\in C(\{0,1\}^{E\vphantom {E^{1^{1}}}})\}$ , where $\pi _{E}\colon \{0,1\}^{G}\to \{0,1\}^{E}$ is given by $\pi _{E}(x)=x|_{E}$ . It follows from equation (2.1) and permutation invariance of $u_{d_{n}}$ that

$$ \begin{align*} \lim_{n\to\infty}\int \alpha_{g}(f)\,d\Theta_{n}-\int f\,d\Theta_{n}=0\quad \text{for all } g\in G, f\in \bigcup_{E\subseteq G \text{ finite}}A_{E}. \end{align*} $$

Thus,

$$ \begin{align*}\int f\,d\Theta_{\omega}=\int \alpha_{g}(f)\,d\Theta_{\omega}\end{align*} $$

for all $g\in G,f\in \bigcup _{E}A_{E}$ . Stone and Weierstrass imply that $\bigcup _{E}A_{E}$ is norm dense in $C(\{0,1\}^{G})$ and so by the Riesz representation theorem, we have shown that $\Theta _{\omega }$ is invariant under the conjugation action of G.

It thus suffices to show that $\Theta _{\omega }$ is supported on the space of subgroups of G. For ${g,h\in G}$ , let

$$ \begin{align*} X_{g,h}&=\{x\in \{0,1\}^{G}:x(gh)\geq x(g)x(h)\},\\ I_{g}&=\{x\in \{0,1\}^{G}:x(g^{-1})=x(g)\}. \end{align*} $$

Then

$$ \begin{align*}\operatorname{\mathrm{Sub}}(G)=\bigcap_{g\in G}I_{g}\cap \bigcap_{g,h\in G}X_{g,h}\cap \{x\in \{0,1\}^{G}:x(e)=1\}.\end{align*} $$

Since G is countable, it suffices to show that $\Theta _{\omega }$ assigns each set in this intersection measure $1$ . By the Portmanteau theorem [Reference Dudley22, Theorem 11.1.1], for each $g,h\in G$ ,

$$ \begin{align*} &\Theta_{\omega}(\{x\in \{0,1\}^{G}:x(gh)\geq x(g)x(h)\})\\ &\quad \geq\lim_{n\to\omega}u_{d_{n}}(\{j:1_{\{j\}}(\sigma_{n}(gh)(j))\geq 1_{\{j\}}(\sigma_{n}(g)(j))1_{\{j\}}(\sigma_{n}(h)(j))\})\\ &\quad\geq \lim_{n\to\omega}u_{d_{n}}(\{j:\sigma_{n}(gh)(j)=\sigma_{n}(g)\sigma_{n}(h)(j)\})=1. \end{align*} $$

The proofs that $\Theta _{\omega }$ assigns measure $1$ to $I_{g}$ and $\{x\in \{0,1\}^{G}:x(e)=1\}$ are similar.

We are thus able to make the following definition.

Definition 2.4. Let G be a countable, discrete group and $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ approximate homomorphisms. Define $\Theta _{n}$ as was done before Definition 2.2. Given $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ , we define $\operatorname {\mathrm {IRS}}(\sigma _{\omega })=\lim _{n\to \omega }\Theta _{n}$ .

We remark on an alternate construction of $\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ . One can take an ultraproduct of the measure spaces $(\{1,\ldots ,d_{n}\},u_{d_{n}})$ to obtain a probability space $({\mathcal L},u_{\omega })$ called the Loeb measure space [Reference Loeb38]. The actions ${\operatorname {Sym}}(d_{n})\,{\curvearrowright }\, \{1,\ldots ,d_{n}\}$ along with the approximate homomorphisms $\sigma _{n}$ will induce a probability measure-preserving action on $({\mathcal L},u_{\omega })$ in a natural way. Under this action, one can show that $\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ is ${\operatorname {Stab}}_{*}(u_{\omega })$ , where ${{\operatorname {Stab}}\colon {\mathcal L}\to \operatorname {\mathrm {Sub}}(G)}$ is given by ${\operatorname {Stab}}(z)=\{g\in G:gz=z\}$ . However, we will not need this fact and thus will not prove it. If $\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ does not depend upon $\omega ,$ then $\lim _{n\to \infty }\Theta _{n}$ exists. In this case, we call $\lim _{n\to \infty }\Theta _{n}$ the stabilizer type of $\sigma _{n}$ . For example, $\lim _{n\to \infty }\Theta _{n}=\delta _{\{1\}}$ if and only if $\sigma _{n}$ is a sofic approximation.

3. Background on orbit equivalence relations and Theorem 1.1

To extend approximate homomorphisms to sofic approximations of relations, we need the following construction (appearing first in [Reference Abért, Glasner and Virág2]) of an action associated to an IRS, say $\Theta $ . The intention is that the action is ‘Bernoulli as possible’ while still having $\Theta $ as its IRS. For technical reasons, the action will not always have $\Theta $ 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 $\Theta \in \operatorname {\mathrm {IRS}}(G)$ . Let X be a compact metrizable space and $\nu $ a Borel probability measure on G. Let

$$ \begin{align*} Y=\{(H,x)\in \operatorname{\mathrm{Sub}}(G)\times X^{G}:x(hg)=x(g) \text{ for all } g\in G,h\in H\}. \end{align*} $$

Observe that Y is a closed subset of $\operatorname {\mathrm {Sub}}(G)\times X^{G}$ . For $H\in \operatorname {\mathrm {Sub}}(G)$ , let $G/H$ be the space of right cosets of H in G and regard $\nu ^{\otimes G/H}$ as a probability measure on $X^{G}$ which is supported on the $x\in X^{G}$ which are constant on right H-cosets. Let $\mu _{\Theta }$ be the measure

$$ \begin{align*}\mu_{\Theta}=\int_{\operatorname{\mathrm{Sub}}(G)}\delta_{H}\otimes \nu^{\otimes G/H}\,d\Theta(H).\end{align*} $$

We sometimes denote $(Y,\mu _{\Theta })$ as $\operatorname {\mathrm {Bern}}(X,\nu ,\Theta )$ . We let $G\,{\curvearrowright }\, X^{G}$ by

$$ \begin{align*} (gx)(a)=x(g^{-1}a)\quad \text{for all } g,a\in G,x\in X^{G}. \end{align*} $$

Note that $G\,{\curvearrowright }\, Y$ by

$$ \begin{align*} g\cdot(H,x)=(gHg^{-1},gx). \end{align*} $$

We call $G\,{\curvearrowright }\, (Y,\mu _{\Theta })$ the $\Theta $ -Bernoulli action with base $(X,\nu )$ .

Suppose that $\nu $ is not a dirac mass, and choose $b\in [0,1)$ so that $\nu (\{x\})\in [0,b]$ for $x\in X$ . Suppose that $H\in \operatorname {\mathrm {Sub}}(G)$ and $F\subseteq G/H$ is finite with $H\in F$ , then by Fubini and Tonelli,

$$ \begin{align*} \nu^{\otimes G/H}(\{x\in X^{G}:x(g_{1})&=x(g_{2}) \text{ for all } c_{1},c_{2}\in G/H, \text{ and all } g_{1}\in c_{1},g_{2}\in c_{2}\}) \\ &\leq \nu^{\otimes G/H}(\{x\in X^{G}:x(g)=x(e) \text{ for all } c\in F, \text{ and all } g\in c\})\\ &\leq b^{|F|}. \end{align*} $$

Since $b<1$ , and the above inequality is true for every finite $F\subseteq G$ , we deduce that if either:

• • $\nu $ is atomless; or

• • H is infinite index in G almost surely,

then for $\mu _{\Theta }$ -almost every $(H,x)$ with $H\ne G$ , we have ${\operatorname {Stab}}((H,x))=H$ . Note that ${\operatorname {Stab}}((G,x))$ =G. So, under either of the above bulleted conditions, the action $G\,{\curvearrowright }\, (Y,\mu _{\Theta })$ has IRS equal to $\Theta .$ This construction first appears in [Reference Abért, Glasner and Virág1, Proposition 13] (see [Reference Creutz and Peterson17, §§3.2–3.3] for the locally compact case, as well as a proof of the fact that if $\Theta $ is ergodic, then we can modify the above construction to get an ergodic action). See [Reference Seward46], [Reference Tucker-Drob48, §5] for further applications of this construction.

The classical Bernoulli shift with base $(X,\nu )$ is the case when $\Theta =\delta _{\{1\}}$ . It is of great importance in ergodic theory, via its connections to probability (it is the sample space for independent and identically distributed X-valued random variables $(\Upsilon _{g})_{g\in G}$ ). It also has many desirable properties such as being mixing ([Reference Petersen43, §2.5] and [Reference Kerr and Li37, §2.3]), complete positive entropy [Reference Kerr35, Reference Rudolph and Weiss45], Koopman representation being an infinite direct sum of the left regular [Reference Kerr and Li37, §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 [Reference Walters50, §4.9], [Reference Petersen43, §§6.4 and 6.5], and [Reference Kerr and Li37, §2.3] for more details and information on the classical Bernoulli shift. For our purposes, we will only need that the $\Theta $ -Bernoulli shift retains a residue of freeness, in that under the above conditions, it has $\Theta $ as its IRS.

We will need the notion of an orbit equivalence relation.

The last item can be recast as follows: define a Borel measure $\overline {\nu }$ on ${\mathcal R}$ by

$$ \begin{align*}\overline{\nu}(B)=\int_{X}|\{y\in [x]_{{\mathcal R}}:(x,y)\in B\}|\,d\nu(x).\end{align*} $$

Then the map $(x,y)\mapsto (y,x)$ preserves the measure if and only if the last item holds. This implies, for example, that if ${\mathcal R}$ is a discrete, probability measure-preserving equivalence relation on $(X,\nu )$ and if $f\in L^{1}({\mathcal R})$ , then

$$ \begin{align*}\int \sum_{y\in [x]_{{\mathcal R}}}f(x,y)\,d\nu(x)=\int \sum_{y\in [x]_{{\mathcal R}}}f(y,x)\,d\nu(x).\end{align*} $$

If G is a countable discrete group and $G\,{\curvearrowright }\, (X,\nu )$ is a probability measure-preserving action, we then have a discrete, probability-measure relation given as the orbit equivalence relation

$$ \begin{align*}{\mathcal R}_{G,X}=\{(x,gx):x\in X,g\in G\}.\end{align*} $$

All discrete, probability measure-preserving equivalence relations arise this way [Reference Feldman and Moore28].

As mentioned before, we will extend approximate homomorphisms to sofic approximations of equivalence relations. To define a sofic approximation of an equivalence relation, we use tracial von Neumann algebras.

For a tracial von Neumann algebra $(M,\tau )$ and $x\in M$ , we set $\|x\|_{2}=\tau (x^{*}x)^{1/2}$ . A simple example of a tracial von Neumann algebra is the following. For $k\in {\mathbb N}$ , define $\operatorname {\mathrm {tr}}\colon M_{k}({\mathbb C})\to {\mathbb C}$ by

$$ \begin{align*}\operatorname{\mathrm{tr}}(\upsilon)=\frac{1}{k}\sum_{j=1}^{k}\upsilon_{jj}.\end{align*} $$

Then $(M_{k}({\mathbb C}),\operatorname {\mathrm {tr}})$ 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 [Reference Bekka8, §Reference Abért, Glasner and Virág2].

Given a discrete, probability measure-preserving equivalence relation ${\mathcal R}$ on $(X,\nu )$ , we let $[{\mathcal R}]$ be the group of all bimeasurable bijections $\gamma \colon X_{0}\to Y_{0}$ , where $X_{0},Y_{0}$ are conull subsets of X and with $\gamma (x)\in [x]_{{\mathcal R}}$ for almost every $x\in X$ . As usual, we identify two such maps if they agree off a set of measure zero. The group $[{\mathcal R}]$ is called the full group of ${\mathcal R}$ . We let $[[{\mathcal R}]]$ be the set of all bimeasurable bijections $\gamma \colon B_{1}\to B_{2}$ (where $B_{1},B_{2}$ are measurable subsets of X) which satisfy that $\gamma (x)\in [x]_{{\mathcal R}}$ for almost every $x\in B_{1}$ . As usual, we identify two such maps if they agree off a set of measure zero. We usually use $\operatorname {\mathrm {dom}}(\gamma ),\operatorname {\mathrm {ran}}(\gamma )$ for $B_{1},B_{2}$ above. We define maps $\vartheta \colon L^{\infty }(X,\nu )\to B(L^{2}({\mathcal R},\overline {\nu }))$ and $\unicode{x3bb} \colon [[{\mathcal R}]]\to \mathcal {U}(L^{2}({\mathcal R},\overline {\nu }))$ by

$$ \begin{align*} (\vartheta(f)\xi)(x,y)&=f(x)\xi(x,y),\\ (\unicode{x3bb}(\gamma)\xi)(x,y)&=1_{\operatorname{\mathrm{ran}}(\gamma)}(x)\xi(\gamma^{-1}(x),y). \end{align*} $$

We define the von Neumann algebra of the equivalence relation to be

$$ \begin{align*} L({\mathcal R})=W^{*}(\unicode{x3bb}([{\mathcal R}])\cup \vartheta(L^{\infty}(X,\nu))).\end{align*} $$

Note that if $B\subseteq X$ is measurable, then we have an element $\operatorname {\mathrm {id}}_{B}\in [[{\mathcal R}]]$ with $\operatorname {\mathrm {dom}}(\operatorname {\mathrm {id}}_{B})=E=\operatorname {\mathrm {ran}}(\operatorname {\mathrm {id}}_{B})$ and $\operatorname {\mathrm {id}}_{B}(x)=x$ for all $x\in B$ . Moreover, $\unicode{x3bb} (\operatorname {\mathrm {id}}_{B})=\vartheta (1_{B})$ . Since simple functions are dense in $L^{\infty }(X,\nu )$ , this implies that

$$ \begin{align*} L({\mathcal R})=W^{*}(\unicode{x3bb}([[{\mathcal R}]])). \end{align*} $$

The von Neumann algebra $L({\mathcal R})$ is equipped with a trace

$$ \begin{align*} \tau(x)={\langle {x1_{\Lambda},1_{\Lambda}} \rangle} \end{align*} $$

where $\Lambda =\{(x,x):x\in X\}$ . We typically identify $L^{\infty }(X,\nu )$ and $[[{\mathcal R}]]$ as subsets of $L({\mathcal R})$ and do not make explicit reference to the maps $\unicode{x3bb} ,\vartheta .$

Another example of a tracial von Neumann algebra is the ultraproduct of tracial von Neumann algebras. Let $\omega $ be a free ultrafilter on $\mathbb {N}$ . Suppose $(N_k, \tau _k)$ are tracial von Neumann algebras. Denote the ultraproduct by

$$ \begin{align*} \prod_{k\to \omega} (N_k,\tau_k)= \big\{(x_k)_{k\in \mathbb{N}}\ |\ \sup_{k} \|x_k\|< \infty\big\}/\big\{(x_k)| \lim_{k\to \omega} \|x_k\|_2=0\big\}. \end{align*} $$

If $(x_{k})_{k}\in \prod _{k}N_{k}$ with $\sup _{k}\|x_{k}\|<+\infty ,$ we use $(x_{k})_{k\to \omega }$ for its image in $\prod _{k\to \omega }(N_{k},\tau _{k})$ . By the proof of [Reference Brown and Ozawa12, Lemma A.9], the ultraproduct is a tracial von Neumann algebra and is equipped with a canonical trace $\tau ((x_n)_{\omega })= \lim _{n\to \omega }\tau _n(x_n)$ .

For a sequence of integers $d_{n}$ , set

$$ \begin{align*}({\mathcal M},\tau_{\omega})=\prod_{n\to\omega}(M_{d_{n}}({\mathbb C}),\operatorname{\mathrm{tr}}).\end{align*} $$

We let

$$ \begin{align*}(L^{\infty}({\mathcal L},u_{\omega}),u_{\omega})=\prod_{n\to\omega}(\ell^{\infty}(d_{n}),u_{d_{n}})\end{align*} $$

and

$$ \begin{align*}{\mathcal S}_{\omega}=\prod_{n\to\omega}({\operatorname{Sym}}(d_{n}),d_{\operatorname{\mathrm{Hamm}}}).\end{align*} $$

View $\ell ^{\infty }(d_{n})\subseteq M_{d_{n}}({\mathbb C})$ by identifying each function with the diagonal matrix whose entries are $(f(1),\ldots ,f(d_{n}))$ , and identify each permutation with its corresponding permutation matrix. In this way, we can identify $L^{\infty }({\mathcal L},u_{\omega })$ as a subalgebra of ${\mathcal M}$ , and ${\mathcal S}_{\omega }$ as a subgroup of the unitary group of ${\mathcal M}$ .

Definition 3.5. Let $(X,\nu ,{\mathcal R})$ be a discrete, probability measure-preserving equivalence relation. We say that ${\mathcal R}$ is sofic if there is a free ultrafilter $\omega $ , a sequence of positive integers $(d_{n})_{n}$ , and maps

$$ \begin{align*} \rho\colon L^{\infty}(X,\nu)\to L^{\infty}({\mathcal L},u_{\omega}) &:=\prod_{n\to\omega}(\ell^{\infty}(d_{n}),u_{d_{n}}),\\ \sigma\colon [{\mathcal R}]\to {\mathcal S}_{\omega}&:=\prod_{n\to\omega}({\operatorname{Sym}}(d_{n}),d_{\operatorname{\mathrm{Hamm}}}) \end{align*} $$

so that:

• • $\rho $ is a normal $*$ -homomorphism;

• • $\sigma $ is a homomorphism;

• • $\tau _{\omega }(\rho (f)\sigma (\gamma ))=\int _{\{x\in X:\gamma (x)=x\}}f\,d\nu $ for every $f\in L^{\infty }(X,\nu ),\gamma \in [{\mathcal R}]$ ;

• • $\sigma (\gamma )\rho (f)\sigma (\gamma )^{-1}=\rho (f\circ \gamma ^{-1})$ for every $f\in L^{\infty }(X,\nu ),\gamma \in [{\mathcal R}]$ .

Since $[{\mathcal R}],L^{\infty }(X,\nu )$ 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 and Lippner (see the proof [Reference Elek and Lippner26, Theorem 2]) or [Reference Păunescu42]; therefore, we do not claim originality for these results.

Proof. Throughout, set $({\mathcal M},\tau _{\omega })=\prod _{k\to \omega }(M_{k}({\mathbb C}),\operatorname {\mathrm {tr}}).$

(i) Let

$$ \begin{align*} M_{0}=\bigg\{\sum_{\gamma\in [{\mathcal R}]}a(\gamma)\gamma:a\colon[{\mathcal R}]\to L^{\infty}(X,\nu) \text{ is finitely supported}\bigg\}, \end{align*} $$

then $M_{0}$ is weak operator topology dense in $L({\mathcal R})$ by definition. Moreover, for all $f_{1},\ldots ,f_{n}\in L^{\infty }(X,\nu ),\gamma _{1},\ldots ,\gamma _{n}\in [{\mathcal R}]$ , we have by the axioms of a sofic approximation:

$$ \begin{align*} \bigg\|\sum_{j=1}^{n}\rho(f_{j})\sigma(\gamma_{j})\bigg\|_{2}^{2} &= \sum_{i,j}\tau_{\omega}(\sigma(\gamma_{j})^{-1}\rho(\overline{f_{j}}f_{k})\sigma(\gamma_{k}))\\ &=\sum_{i,j}\tau_{\omega}(\rho((\overline{f_{j}}f_{k})\circ \gamma_{j})\sigma(\gamma_{j}^{-1}\gamma_{k})) \\ &=\sum_{i,j}\int_{\{x:\gamma_{j}^{-1}(\gamma_{k}(x))=x\}}(\overline{f_{j}}f_{k})\circ \gamma_{j}\,d\nu\\&=\sum_{i,j}\tau([(\overline{f_{j}}f_{k})\circ \gamma_{j}\gamma_{j}^{-1}]\unicode{x3bb}(\gamma_{k}))\\ &=\sum_{i,j}\tau(\unicode{x3bb}(\gamma_{j})^{-1}\overline{f_{j}}f_{k}\gamma\unicode{x3bb}(\gamma_{k}))\\ &=\bigg\|\sum_{j=1}^{n}f_{j}\unicode{x3bb}(\gamma_{j})\bigg\|_{2}^{2}. \end{align*} $$

Since $\|\cdot \|_{2}$ is a norm, the above calculation implies that $\sum _{j}\rho (f_{j})\sigma (\gamma _{j})=0$ if and only if $\sum _{j}f_{j}\unicode{x3bb} (\gamma _{j)}=0$ . This implies that the map $\pi _{0}\colon M_{0}\to \prod _{k\to \omega }(M_{k}({\mathbb C}),\operatorname {\mathrm {tr}})$ given by

$$ \begin{align*}\pi_{0}\bigg(\sum_{\gamma}f(\gamma)\unicode{x3bb}(\gamma)\bigg) = \sum_{\gamma}\rho(f(\gamma))\sigma(\gamma)\end{align*} $$

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 $\pi _{0}$ has a unique extension to trace-preserving normal $*$ -homomorphism $\pi $ .

(ii) This is an exercise in understanding the definitions.

(iii) Let $A=\operatorname {\mathrm {span}}(D)$ so that A is a weak operator topology dense $*$ -subalgebra of $L^{\infty }(X,\nu )$ . Let

$$ \begin{align*} M_{0}=\bigg\{\sum_{g\in G}a(g)\unicode{x3bb}(g):a\in c_{c}(G,A)\bigg\}, \end{align*} $$

then $M_{0}$ is $*$ -subalgebra of $L({\mathcal R})$ . As in item (i), we know there is a unique function $\pi _{0}\colon M_{0}\to {\mathcal M}$ satisfying

$$ \begin{align*} \pi_{0}\bigg(\sum_{g\in G}a(g)\unicode{x3bb}(g)\bigg)=\sum_{g\in G}\rho_{0}(a(g))\sigma_{0}(g)\quad \text{for all } a\in c_{c}(G,A). \end{align*} $$

Our hypothesis implies that $\pi _{0}$ 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 $L({\mathcal R})$ . Let M be the weak operator topology closure of $M_{0}$ . Then

$$ \begin{align*} M\supseteq \overline{A}^{\mathrm{weak}^{*}}=L^{\infty}(X,\nu). \end{align*} $$

Since $Gx=[x]_{{\mathcal R}}$ for almost every $x\in X$ , given $\gamma \in [[{\mathcal R}]]$ , we may find (not necessarily unique) disjoint sets $(B_{g})_{g\in G}$ so that $B_{g}\subseteq \{x\in \operatorname {\mathrm {dom}}(\gamma ):\gamma (x)=gx\}$ and with

$$ \begin{align*}\nu\bigg(\operatorname{\mathrm{dom}}(\gamma)\Delta \bigsqcup_{g\in G}B_{g}\bigg)=0.\end{align*} $$

We leave it as an exercise to check that

$$ \begin{align*}\unicode{x3bb}(\gamma)=\sum_{g\in G}1_{B_{g^{-1}}}\unicode{x3bb}(g)\end{align*} $$

with the sum converging in the strong operator topology. Since we have already shown that $L^{\infty }(X,\nu )\subseteq M$ , it follows that $[[{\mathcal R}]]\subseteq M$ . Hence,

$$ \begin{align*}L({\mathcal R})=W^{*}([[{\mathcal R}]])\subseteq M.\end{align*} $$

(iv) Let $D=\{1_{E}:E\in {\mathcal A}\}$ , then $\operatorname {\mathrm {span}}(D)$ is weak $^{*}$ -dense in $L^{\infty }(X)$ since ${\mathcal A}$ is generating. Now apply item (iii).

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 $\pi \colon L({\mathcal R})\to \prod _{k\to \omega }(M_{k}({\mathbb C}),\tau )$ satisfying the hypotheses of item (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 $\rho _{n}\colon {\mathcal A}\to \mathcal {P}(\{1,\ldots ,d_{n}\})$ and $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ so that:

• • $\sigma _{n}$ is an asymptotic homomorphism;

• • $u_{d_{n}}(\rho _{n}(B_{1}\cap B_{2})\Delta (\rho _{n}(B_{1})\cap \rho _{n}(B_{2})))\to _{n\to \infty }0$ for all $B_{1},B_{2}\in {\mathcal A}$ ;

• • $u_{d_{n}}(\{j\in \rho _{n}(B):\sigma _{n}(g)(j)=j\})\to _{n\to \infty } \nu (\{x\in E:gx=x\})$ for all $B\in {\mathcal A}$ , $g\in G$ ;

• • $u_{d_{n}}(\rho _{n}(gB)\Delta (\sigma _{n}(g)\rho _{n}(B)\sigma _{n}(g)^{-1}))\to _{n\to \infty }0$ for all $g\in G$ , $B\in {\mathcal A}$ .

This is sometimes taken as the definition of soficity. Our definition has the advantage of being canonical and not requiring a choice of $G,{\mathcal A}$ . 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 $\rho _{n}\colon D\to \ell ^{\infty }(d_{n})$ , $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ so that:

• • $\|\rho _{n}(f_{1}f_{2})-\rho _{n}(f_{1})\rho _{n}(f_{2})\|_{2}\to _{n\to \infty }0$ for all $f_{1},f_{2}\in D$ ;

• • $\sigma _{n}$ is an asymptotic homomorphism;

• • ${1}/{d_{n}} \sum _{j:\sigma _{n}(g)(j)=j}\rho _{n}(f)(j)\to _{n\to \infty }\int _{\{x\in X:gx=x\}}f\,d\nu $ for all $f\in D,g\in G$ ;

• • $\|\rho _{n}(f)\circ \sigma _{n}(g)^{-1}-\rho _{n}(\alpha _{g}(f))\|_{2}\to _{n\to \infty } 0$ for every $f\in D,g\in G$ .

Here, the $\|\cdot \|_{2}$ -norms are with respect to the uniform measure and $\alpha _{g}(f)(x)=f(g^{-1}x)$ . 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. Proof of Theorem 3.7

We now proceed to prove Theorem 3.7. We remark that one can use [Reference Tucker-Drob48, 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 $\Theta $ , the induced action on the Loeb measure space $G\,{\curvearrowright }\, ({\mathcal L},u_{\omega })$ also has IRS $\Theta $ . 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 $\Theta $ . Being an action on a standard probability space, such a factor is weakly contained in a $\Theta $ -Bernoulli shift by [Reference Tucker-Drob48, Theorem 1.5]. This implies that the $\Theta $ -Bernoulli shift action $G\,{\curvearrowright }\, (X_{\Theta },\mu _{\Theta })$ with base $[0,1]$ (equipped with Lebesgue measure) is weakly contained in $G\,{\curvearrowright }\, ({\mathcal L},u_{\omega })$ . Since $G\,{\curvearrowright }\, ({\mathcal L},u_{\omega })$ is an ultraproduct action, if $G\,{\curvearrowright }\, (X_{\Theta },\mu _{\Theta })$ is weakly contained in $G\,{\curvearrowright }\, ({\mathcal L},u_{\omega })$ , it must actually be a factor of this action. Putting this altogether shows that any sequences of approximate homomorphisms extend to a sofic approximation of the $\Theta $ -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 [Reference Bowen10, Theorem 8.1]. Bowen’s purpose in [Reference Bowen10] 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 [Reference Seward46, Theorem 9.1] which also says, in some sense, that the $\Theta $ -Bernoulli shift with base $(X,\nu )$ is the ‘largest entropy’ action whose IRS is $\Theta $ and which is generated by the translates of an X-valued random variable with distribution $\nu $ .

For ease of notation, if X is a compact, metrizable space, $\nu \in \operatorname {\mathrm {Prob}}(X)$ , and ${f\colon X\to {\mathbb C}}$ is bounded and Borel, we often use $\nu (f)$ for $\int f\,d\nu $ . Suppose I is a set and $(X_{i})_{i\in I}$ are compact Hausdorff spaces. If $E\subseteq I$ is finite and $(f_{i})_{i\in E}\in \prod _{i\in E}C(X_{i}),$ we define $\bigotimes _{i\in E}f_{i}\in C(\prod _{i\in I}X_{i})$ by

$$ \begin{align*} \bigg(\bigotimes_{i\in E}f_{i}\bigg)(x)=\prod_{i\in E}f_{i}(x_{i})\quad \text{if } x=(x_{i})_{i\in I}\in \prod_{i\in I}X_{i}. \end{align*} $$

For a finite $F\subseteq G$ , we define ${\mathcal E}_{F}\in C(\{0,1\}^{G}$ ) by

$$ \begin{align*} {\mathcal E}_{F}(a)=\prod_{g\in F}a(g). \end{align*} $$

Viewing $\operatorname {\mathrm {Sub}}(G)\subseteq \{0,1\}^{G}$ , we have ${\mathcal E}_{F}|_{\operatorname {\mathrm {Sub}}(G)}$ is the indicator function of $\{H:F\subseteq H\}$ . If Y is a compact, metrizable space, we often slightly abuse notation and regard ${{\mathcal E}_{F}\in C(\{0,1\}^{G}\times Y)}$ via the embedding $C(\{0,1\}^{G})\to C(\{0,1\}^{G}\times Y)$ given by

$$ \begin{align*}f\mapsto ((a,y)\mapsto f(a)).\end{align*} $$

Lemma 4.1. Let $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ be a sequence of asymptotic homomorphisms and $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ . Set $\Theta =\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ . Let $\mu _{\Theta }$ be the $\Theta $ -Bernoulli measure on $\operatorname {\mathrm {Sub}}(G)\times [0,1]^{G}$ as in Definition 3.1. View $\operatorname {\mathrm {Sub}}(G)\times [0,1]^{G}\subseteq (\{0,1\}\times [0,1])^{G}$ by identifying each subgroup with its indicator function. For $x\in [0,1]^{d_{n}}$ , define $\phi _{x}\colon [d_{n}]\to (\{0,1\}\times [0,1])^{G}$  by

$$ \begin{align*}\phi_{x}(j)(g)=(1_{\{j\}}(\sigma_{n}(g)(j)),x(\sigma_{n}(g)^{-1}(j))).\end{align*} $$

Then, we have the following items.

1. (i) in the weak $^{*}$ -topology,

   $$ \begin{align*} \lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})\,dx = \mu_{\Theta}. \end{align*} $$

2. (ii) Given $f\in C(\{0,1\}^{G}\times [0,1]^{G})$ , set $J_{n,f}=\int _{[0,1]^{d_{n}}}(\phi _{x})_{*}(u_{d_{n}})(f)\,dx$ . Then

   $$ \begin{align*} \lim_{n\to\infty}\int_{[0,1]^{d_{n}}}|(\phi_{x})_{*}(u_{d_{n}})(f)-J_{n,f}|^{2}\,dx=0. \end{align*} $$

3. (iii) For $g\in G$ , define $\alpha _{g}\in B(C((\{0,1\}\times [0,1])^{G}))$ by $(\alpha _{g}(f))(H,x)= f(g^{-1}Hg, g^{-1}x)$ . Then for any $f\in C(\{0,1\}\times [0,1])^{G})$ , $g\in G$ ,

   $$ \begin{align*}\lim_{n\to\infty}\sup_{x\in [0,1]^{d_{n}}}\|f\circ \phi_{x}\circ \sigma_{n}(g)^{-1}-\alpha_{g}(f)\circ \phi_{x}\|_{\ell^{2}(u_{d_{n}})}=0.\end{align*} $$

Proof. Throughout, we use $G/H$ for the space of right cosets of H in G. For $F\subseteq G$ finite, we set

$$ \begin{align*}\varsigma_{F,n}=\bigcap_{g\in F}\{j:\sigma_{n}(g)(j)=j\}.\end{align*} $$

(i) By the Riesz representation theorem, we can define $\eta \in \operatorname {\mathrm {Prob}}((\{0,1\}\times [0,1])^{G})$ by

$$ \begin{align*} \eta(f)=\lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx\quad \text{for all } f\in C((\{0,1\}\times [0,1])^{G}). \end{align*} $$

Let

$$ \begin{align*} \digamma=\bigg\{\bigg(\bigotimes_{g\in E}f_{g}\bigg){\mathcal E}_{F}:E, F\subseteq G \text{ are finite}, (f_{g})_{g\in E}\in C([0,1])^{E},f_{g}\geq 0 \text{ for all } g\in E\bigg\}. \end{align*} $$

By Stone and Weierstrass, $\operatorname {\mathrm {span}}(\digamma )$ is norm dense in $C((\{0,1\}\times [0,1])^{G})$ , and by the Riesz representation theorem to show that $\eta =\mu _{\Theta }$ , it suffices to show that they have the same integral against any element of $\digamma $ . Fix $f=(\bigotimes _{g\in E}f_{g}){\mathcal E}_{F}\in \digamma $ . Let $\Omega _{n}$ be the set of $j\in [d_{n}]$ so that for all $1\leq k\leq (2023)!$ and for all $g_{1},\ldots ,g_{k}\in E$ and all $s_{1},\ldots ,s_{k}\in \{-1,1\}$ , we have that

$$ \begin{align*}\sigma_{n}(g_{1}^{s_{1}}\cdots g_{k}^{s_{k}})(j)=\sigma_{n}(g_{1})^{s_{1}}\cdots \sigma_{n}(g_{k})^{s_{k}}(j).\end{align*} $$

Since $\sigma _{n}$ is an approximate homomorphism, $u_{d_{n}}(\Omega _{n})\to 1$ . Hence,

$$ \begin{align*} \int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx&=\frac{1}{d_{n}}\sum_{j\in \varsigma_{F,n}}\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx\\ &=u_{d_{n}}(\Omega_{n}^{c})\frac{1}{|\Omega_{n}|^{c}}\sum_{j\in \varsigma_{F,n}\setminus \Omega_{n}}\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx\\ &\quad +u_{d_{n}}(\Omega_{n})\frac{1}{|\Omega_{n}|}\!\sum_{j\in \varsigma_{F,n}\cap \Omega_{n}}\!\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx. \end{align*} $$

Since $u_{d_{n}}(\Omega _{n})\to 1$ and the $f_{g}$ are bounded, the first term tends to zero. It thus remains to analyze the second term. For $j\in \Omega _{n}$ ,

$$ \begin{align*} &\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx=\prod_{s\in \sigma_{n}(E^{-1})(j)}\int_{0}^{1}\prod_{g\in E:\sigma_{n}(g^{-1})(j)=s}f_{g}(x)\,dx \\ &\quad =\prod_{g\in E}\bigg(\int_{0}^{1}\prod_{k\in E:\sigma_{n}(k^{-1})(j)=\sigma_{n}(g^{-1})(j)}f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:\sigma_{n}(a^{-1})(j)=\sigma_{n}(g^{-1})(j)\}|}}, \end{align*} $$

where in the last step, we use that $f_{g}\geq 0$ to make sense of the fractional power (which is there to account for over-counting). Because $j\in \Omega _{n}$ ,

$$ \begin{align*} &\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx\\ &\quad =\prod_{g\in E}\bigg(\int_{0}^{1}\prod_{k\in E}1_{\{j\}}(\sigma_{n}(gk^{-1})(j))f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:\sigma_{n}(ga^{-1})(j)=j\}|}}. \end{align*} $$

Since $u_{d_{n}}(\Omega _{n})\to 1$ , we have altogether shown that

$$ \begin{align*} &\lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx\\ &\quad =\lim_{n\to\omega}\frac{1}{d_{n}}\sum_{j\in [d_{n}]}\prod_{b\in F}1_{\{j\}}(\sigma_{n}(b)(j))\prod_{g\in E}\\ &\qquad\times \bigg(\int_{0}^{1}\prod_{k\in E}1_{\{j\}}(\sigma_{n}(gk^{-1})(j))f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:\sigma_{n}(ga^{-1})(j)=j\}|}}. \end{align*} $$

It is direct to verify that the function $\Phi _{f}\colon \{0,1\}^{G}\to [0,+\infty )$ given by

$$ \begin{align*}\Phi_{f}(y)=\bigg(\prod_{b\in F}y(b)\bigg)\prod_{g\in E}\bigg(\int_{0}^{1}\prod_{k\in E}y(gk^{-1})f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:y(ga^{-1})=1\}|}}\end{align*} $$

is continuous (in fact, it is locally constant). The definition of $\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ thus proves that

$$ \begin{align*}\lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx= \int \Phi_{f}(H)\,d\Theta(H).\end{align*} $$

By definition,

$$ \begin{align*}\bigg(y\mapsto \prod_{b\in F}y(b)\bigg)={\mathcal E}_{F},\end{align*} $$

moreover, if $H\in \operatorname {\mathrm {Sub}}(G)$ , then $1_{H}(ga^{-1})=1$ if and only if $Ha=Hg$ . Thus, viewing $\operatorname {\mathrm {Sub}}(G)\subseteq \{0,1\}^{G}$ , we have

$$ \begin{align*} \Phi_{f}(H)\!=\!{\mathcal E}_{F}(H)\!\prod_{g\in E}\!\bigg(\int_{0}^{1}\prod_{k\in E}1_{H}(gk^{-1})f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:Ha=Hg\}|}}\quad \text{for all } H\!\in\! \operatorname{\mathrm{Sub}}(G). \end{align*} $$

So

$$ \begin{align*} &\lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx\\ &\quad =\int_{\operatorname{\mathrm{Sub}}(G)}{\mathcal E}_{F}(H)\prod_{g\in E}\bigg(\int_{0}^{1}\prod_{k\in E}1_{H}(gk^{-1})f_{k}(x)\,dx\bigg)^{{1}/{|\{a\in E:Ha=Hg\}|}}\,d\Theta(H). \end{align*} $$

Note that

$$ \begin{align*}\prod_{c\in HE/H}\bigg(\!\kern1.2pt\int_{0}^{1}\prod_{g\in c\cap E}f_{g}(x)\,dx\!\bigg) = \prod_{g\in E}\bigg(\!\int_{0}^{1}\prod_{k\in E}1_{H}(gk^{-1})f_{k}(x)(x)\,dx\!\bigg)^{{1}/{|\{a\in E:Ha=Hg\}|}},\end{align*} $$

with the last step following because the inverse image of g under the map $E\mapsto G/H, g\mapsto Hg$ has cardinality $|\{a\in E:Ha=Hg\}|.$ Thus,

$$ \begin{align*} \lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx=\int_{\operatorname{\mathrm{Sub}}(G)}{\mathcal E}_{F}(H)\prod_{c\in HE/H}\bigg(\!\int_{0}^{1}\prod_{g\in c\cap E}f_{g}(x)\,dx\bigg)\,d\Theta(H). \end{align*} $$

Note that for $H\in \operatorname {\mathrm {Sub}}(G)$ , we have that

$$ \begin{align*}\int_{[0,1]^{G}}\prod_{g\in E}f_{g}(x(g))\,dm^{\otimes G/H}(x)=\prod_{c\in HE/H}\bigg(\int_{0}^{1}\prod_{g\in c\cap E}f_{g}(x)\,dx\bigg),\end{align*} $$

where we view $m^{\otimes G/H}$ as a probability measure on $[0,1]^{G}$ that is supported on the $x\in [0,1]^{G}$ that are constant on right H-cosets. So

$$ \begin{align*} \int\! {\mathcal E}_{F}(H)\!\prod_{g\in E}f_{g}(x(g))\,d\mu_{\Theta}(H,x)=\int_{\operatorname{\mathrm{Sub}}(G)}\!{\mathcal E}_{F}(H)\!\prod_{c\in HE/H}\!\bigg(\!\int_{0}^{1}\!\prod_{g\in c\cap E}f_{g}(x)\,dx\!\bigg)\,d\Theta(H). \end{align*} $$

Altogether, this shows that

$$ \begin{align*} \lim_{n\to\omega}\int_{[0,1]^{d_{n}}}(\phi_{x})_{*}(u_{d_{n}})(f)\,dx&=\int {\mathcal E}_{F}(H)\prod_{g\in E}f_{g}(x(g))\,d\mu_{\Theta}(H,x)\\ &=\int f\,d\mu_{\Theta}, \end{align*} $$

as desired.

(ii) Define $T_{n}\colon C((\{0,1\}\times [0,1])^{G})\to L^{2}([0,1]^{d_{n}})$ by

$$ \begin{align*}(T_{n}\zeta)(x)=(\phi_{x})_{*}(u_{d_{n}})(\zeta)-J_{n,\zeta}.\end{align*} $$

Giving $C((\{0,1\}\times [0,1])^{G})$ the supremum norm, we have the operator norm of $T_{n}$ satisfies $\|T_{n}\|\leq 2$ . This uniform estimate implies that $\{\zeta \in C((\{0,1\}\times [0,1])^{G}):\|T_{n}\zeta \|_{2}\to _{n\to \omega }0\}$ is a closed, linear subspace. So as in item (i), it suffices to verify the desired statement for $f=(\bigotimes _{g\in E}f_{g}){\mathcal E}_{F}\in B$ , where $(f_{g})_{g\in E}\in C([0,1])^{G}$ and $E,F\subseteq G$ are finite. By direct computation,

(4.1) $$ \begin{align} \int_{[0,1]^{d_{n}}}|(\phi_{x})_{*}(u_{d_{n}})(f)-J_{n,f}|^{2}\,dx = \int_{[0,1]^{d_{n}}}|(\phi_{x})_{*}(u_{d_{n}})|^{2}\,dx-|J_{n,f}|^{2}. \end{align} $$

Moreover,

$$ \begin{align*} &\int_{[0,1]^{d_{n}}}|(\phi_{x})_{*}(u_{d_{n}})|^{2}\,dx\\ &\quad =\frac{1}{d_{n}^{2}}\sum_{j,k\in \varsigma_{F,n}}\int_{[0,1]^{d_{n}}}\prod_{g,h\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\overline{f_{h}(x(\sigma_{n}(h)^{-1}(j)))}\,dx. \end{align*} $$

We can rewrite this as

(4.2) $$ \begin{align} &\frac{1}{d_{n}^{2}}\sum_{\substack{j,k\in \varsigma_{F,n},\\ \sigma_{n}(E)^{-1}(j)\cap \sigma_{n}(E)^{-1}(k)\ne \varnothing}}\int_{[0,1]^{d_{n}}}\prod_{g,h\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\overline{f_{h}(x(\sigma_{n}(h)^{-1}(k)))}\,dx\notag\\ &\quad +\frac{1}{d_{n}^{2}}\sum_{\substack{j,k\in \varsigma_{F,n},\\ \sigma_{n}(E)^{-1}(j)\cap \sigma_{n}(E)^{-1}(k)= \varnothing}}\int_{[0,1]^{d_{n}}}\prod_{g,h\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\overline{f_{h}(x(\sigma_{n}(h)^{-1}(k)))}\,dx. \end{align} $$

For a fixed j, the set of k for which $\sigma _{n}(E)^{-1}(j)\cap \sigma _{n}(E)^{-1}(k)\ne \varnothing $ has cardinality at most $|E|^{2}$ . So the first term is bounded by

$$ \begin{align*}\frac{1}{d_{n}}|E|^{2}\prod_{g\in E}\|f_{g}\|_{\infty}^{2}\to_{n\to\omega}0.\end{align*} $$

For $n\in {\mathbb N}$ , define $\Psi _{n,f}\colon \{1,\ldots ,d_{n}\}\to {\mathbb R}$ by

$$ \begin{align*} \Psi_{n,f}(j)=\int_{[0,1]^{d_{n}}}\prod_{g\in E}f_{g}(x(\sigma_{n}(g)^{-1}(j)))\,dx. \end{align*} $$

The second term in equation (4.2) is

$$ \begin{align*} \frac{1}{d_{n}^{2}}\sum_{\substack{j,k\in \varsigma_{F,n},\\ \sigma_{n}(E)^{-1}(j)\cap \sigma_{n}(E)^{-1}(k)= \varnothing}}\Psi_{n,f}(j)\overline{\Psi_{n,f}(k)}. \end{align*} $$

Thus,

(4.3) $$ \begin{align} \lim_{n\to\omega}\int_{[0,1]^{d_{n}}}|(\phi_{x})_{*}(u_{d_{n}})|^{2}\,dx=\lim_{n\to\omega} \frac{1}{d_{n}^{2}}\sum_{\substack{j,k\in \varsigma_{F,n},\\ \sigma_{n}(E)^{-1}(j)\cap \sigma_{n}(E)^{-1}(k)=\varnothing}}\Psi_{n,f}(j)\overline{\Psi_{n,f}(k)}. \end{align} $$

By another direct computation,

$$ \begin{align*}|J_{n,f}|^{2}=\frac{1}{d_{n}^{2}}\sum_{j,k\in \varsigma_{F,n}}\Psi_{n,f}(j)\overline{\Psi_{n,f}(k)}.\end{align*} $$

By the same estimates as above,

$$ \begin{align*}\lim_{n\to\omega}|J_{n,f}|^{2}=\lim_{n\to\omega}\frac{1}{d_{n}^{2}}\sum_{\substack{j,k\in \varsigma_{F,n},\\ \sigma_{n}(E)^{-1}(j)\cap \sigma_{n}(E)^{-1}(k)=\varnothing}}\Psi_{n,f}(j)\overline{\Psi_{n,f}(k)}.\end{align*} $$

Combining this with equations (4.1), (4.3) shows that

$$ \begin{align*}\lim_{n\to\omega}\int_{[0,1]^{d_{n}}}| (\phi_{x})_{*}(u_{d_{n}})(f)-J_{n,f}|^{2}\,dx=0.\end{align*} $$

Since this is true for every free ultrafilter, we have proven item (ii).

(iii) Since f is continuous, we can find a $C\geq 0$ with $|f|\leq C$ . View $\{0,1\}\times [0,1]\subseteq {\mathbb R}^{2}$ and give ${\mathbb R}^{2}$ the norm $\|(t,s)\|=\sqrt {t^{2}+s^{2}}$ . Given $\varepsilon>0$ , compactness of $(\{0,1\}\times [0,1])^{G}$ and continuity of f imply that we may find a finite $E\subseteq G$ and a $\delta>0$ so that if $x,y\in (\{0,1\}\times [0,1])^{G}$ and

$$ \begin{align*} \|x(g)-y(g)\|<\delta\quad \text{for all } g\in E, \end{align*} $$

then

$$ \begin{align*}|f(x)-f(y)|<\varepsilon.\end{align*} $$

Then, for any $x\in [0,1]^{d_{n}}$ , we have

$$ \begin{align*}\|f\circ \phi_{x}\circ \sigma_{n}(g)^{-1}&-\alpha_{g}(f)\circ \phi_{x}\|_{\ell^{2}(u_{d_{n}})}^{2}\\ &\leq\varepsilon^{2}+C^{2}u_{d_{n}}(\{j\in [d_{n}]:g^{-1}\phi_{x}(j)|_{E}\ne \phi_{x}(\sigma_{n}(g)^{-1}(j))|_{E}\})\\ &\leq \varepsilon^{2}+C^{2}u_{d_{n}}\bigg(\bigcup_{h\in E}\{j:\sigma_{n}(gh)^{-1}(j)\ne \sigma_{n}(h)^{-1}\sigma_{n}(g)^{-1}(j)\}\bigg). \end{align*} $$

Since $\sigma _{n}$ is an asymptotic homomorphism, the second term tends to zero as $n\to \infty $ .

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 $x\in [0,1]^{d_{n}}$ will produce a sofic approximation of the appropriate equivalence relation.

Proof of Theorem 3.7

For notation, define $S\colon [d_{n}]\to \{0,1\}^{G}$ by $S(j)(g)= 1_{\{j\}}(\sigma _{n}(g)(j)).$ Set $\Theta _{n}=S_{*}(u_{d_{n}}).$ For $n\in {\mathbb N},f\in C((\{0,1\}\times [0,1])^{G})$ , let $J_{n,f}=\int _{[0,1]^{d_{n}}} (\phi _{x})_{*}(u_{d_{n}})(f)\,dx$ . Fix a countable, dense set $D\subseteq C((\{0,1\}\times [0,1])^{G})$ which is closed under products and is G-invariant. Write $D=\bigcup _{k=1}^{\infty }D_{k}$ and $G=\bigcup _{n=1}^{\infty }F_{k}$ where $D_{k},F_{k}$ are increasing sequences of finite sets. For $k\in {\mathbb N}$ , let $L_{k}$ be the set of $n\in {\mathbb N}$ so that:

• • $n\geq k$ ;

• • $\sum _{f\in D_{k},g\in F_{k}}|\mu _{\Theta }(f{\mathcal E}_{g})-J_{n,f{\mathcal E}_{g}}|<2^{-k}$ ;

• • $\sum _{f\in D_{k},g\in F_{k}}(\int _{[0,1]^{d_{n}}}|(\phi _{x})_{*}(u_{d_{n}})(f{\mathcal E}_{g})-J_{n,f{\mathcal E}_{g}}|^{2}\,dx)^{1/2}<2^{-k}$ ;

• • $\sup _{x\in [0,1]^{d_{n}}}\sum _{g\in F_{k},f\in D_{k}}\|f\circ \phi _{x}\circ \sigma _{n}(g)^{-1}-\alpha _{g}(f)\circ \phi _{x}\|_{\ell ^{2}(u_{d_{n}})}<2^{-k}.$

Then $L_{k}$ is a decreasing sequence of sets and Lemma 4.1 implies that $L_{k}\in \omega $ for all $k\in {\mathbb N}$ , and $\bigcap _{k}L_{k}=\varnothing $ by the first bullet point. Set $L_{0}={\mathbb N}\setminus L_{1}$ . For $n\in {\mathbb N}$ , let $k(n)$ be such that $n\in L_{k(n)}\setminus L_{k(n)+1}$ . For $n\in {\mathbb N}$ , let $\Omega _{n}$ be the set of $x\in [0,1]^{d_{n}}$ so that

(4.4) $$ \begin{align}\sum_{f\in D_{k(n)},g\in F_{k(n)}}|(\phi_{x})_{*}(u_{d_{n}})(f{\mathcal E}_{g})-J_{n,f{\mathcal E}_{g}}|<2^{-k/2}. \end{align} $$

Then for $n\in L_{k}$ , we have $m^{\otimes d_{n}}(\Omega _{n}^{c})\leq 2^{-k/2}$ . Hence, if $k\geq (2023)!$ and $n\in L_{k}$ , we may choose an $x_{n}\in \Omega _{n}$ . Let $x_{n}$ be defined arbitrarily for $n\in {\mathbb N}\setminus L_{(2023)!}$ . Define

$$ \begin{align*}\rho_{0}\colon D\to \prod_{k\to\omega}(\ell^{\infty}(d_{n}),u_{d_{n}}))\end{align*} $$

by $\rho _{0}(f)=(f\circ \phi _{x_{k(n)}})_{n\to \omega }$ . Then $\rho _{0}$ preserves products. Recall that we view $\ell ^{\infty }(d_{n})$ as a subalgebra of $M_{d_{n}}({\mathbb C})$ by identifying each function with the corresponding diagonal matrix. Let $f\in D,g\in G$ . By the fact that $\sigma _{n}$ is an asymptotic homomorphism and the fourth bullet point above, we know that

$$ \begin{align*}\sigma_{\omega}(g)\rho_{0}(f)\sigma_{\omega}(g)^{-1}=\rho_{0}(\alpha_{g}(f)).\end{align*} $$

By our choice of $x_{n}$ ,

$$ \begin{align*} \tau_{\omega}(\rho_{0}(f)\sigma_{\omega}(g))=\lim_{n\to\omega}\operatorname{\mathrm{tr}}(f\circ \phi_{x_{k(n)}}\sigma_{n}(g))&=\lim_{n\to\omega}\frac{1}{d_{n}}\sum_{j}f(\phi_{x_{k(n)}}(j))1_{\{j\}}(\sigma_{n}(g)(j))\\ &=\lim_{n\to\omega}J_{n,f{\mathcal E}_{g}}, \end{align*} $$

where in the last step, we use equation (4.4). By Lemma 4.1, this last limit is

$$ \begin{align*}\mu_{\Theta}(f{\mathcal E}_{g})=\int_{\{(H,x):g\in H\}}f(H,x)\,d\mu_{\Theta}(H,x).\end{align*} $$

Since $([0,1],m)$ is atomless and G is countable,

$$ \begin{align*} \mu_{\Theta}\bigg(\bigcup_{g_{1},g_{2}\in G}\{(H,x):Hg_{1}\ne Hg_{2} \text{ and } x(g_{1})=x(g_{2})\}\bigg)=0. \end{align*} $$

Thus, for every $g\in G$ ,

$$ \begin{align*} \mu_{\Theta}(\{(H,x):(gHg^{-1},gx)=(H,x)\}\Delta\{(H,x):g\in H\})=0. \end{align*} $$

So

$$ \begin{align*} \tau_{\omega}(\rho_{0}(f)\sigma_{\omega}(g))=\int_{\{(H,x):(gHg^{-1},gx)=(H,x)\}}f\,d\mu_{\Theta}. \end{align*} $$

Thus, Proposition 3.6(iii) implies that there is a unique sofic approximation ${\pi \colon L({\mathcal R})\to \prod _{n\to \omega }(M_{d_{n}}({\mathbb C}),\operatorname {\mathrm {tr}})}$ so that $\pi \circ \Xi =\sigma _{\omega }$ and $\pi |_{D}=\rho _{0}$ . Set $\rho =\pi |_{L^{\infty }(X)}$ and $\widehat {\sigma }=\pi |_{[{\mathcal R}]}$ . By Proposition 3.6(ii), we have that $(\rho ,\widehat {\sigma })$ is a sofic approximation. The fact that $\widehat {\sigma }\circ \Xi =\sigma _{\omega }$ is true by construction.

5. 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 [Reference Newman and Sohler39] (see [Reference Elek25, Theorem 5] for a statement of the Newman–Sohler theorem which is closer to our language).

Proof. Let $\Theta =\operatorname {\mathrm {IRS}}(\sigma _{\omega })=\operatorname {\mathrm {IRS}}(\psi _{\omega })$ and let ${\mathcal R}$ be the orbit equivalence relation of the $\Theta $ -Bernoulli shift with base $([0,1],m)$ . Let $\Xi $ be given as in Theorem 3.7. By Theorem 3.7, we can find sofic approximations $(\rho _{j},\widehat {\sigma }_{j}),j=1,2$ so that $\widehat {\sigma }_{1}\circ \Xi =\sigma _{\omega }$ , $\widehat {\sigma }_{2}\circ \Xi =\psi _{\omega }$ . Since G is amenable, we know by [Reference Ornstein and Weiss40], [Reference Ornstein and Weiss41, II §3] (see also [Reference Connes, Feldman and Weiss15]) that ${\mathcal R}$ is hyperfinite. Thus, by [Reference Păunescu42, Proposition 1.20], there is a $\chi \in {\mathcal S}_{\omega }$ with

$$ \begin{align*} \chi\widehat{\sigma}_{1}(\alpha)\chi^{-1}=\widehat{\sigma}_{2}(\alpha)\quad \text{for all } \alpha\in [{\mathcal R}]. \end{align*} $$

In particular,

$$ \begin{align*} \chi \sigma_{\omega}(g)\chi^{-1}=\chi\widehat{\sigma}_{1}(\Xi(g)) \chi^{-1}=\widehat{\sigma}_{2}(\Xi(g))=\psi_{\omega}(g)\end{align*} $$

for all $g\in G$ .

We remark that Theorem 5.1 recovers the result of Kerr and Li [Reference Kerr and Li36, Lemma 4.5] and Elek and Szabo [Reference Elek and Szabó27] on uniqueness of sofic approximations of amenable groups up to asymptotic conjugacy. Indeed, sofic approximations correspond to the cases ${\operatorname {\mathrm {IRS}}(\sigma _{\omega })=\operatorname {\mathrm {IRS}}(\psi _{\omega })=\{1\}}$ . Another reformulation of this result is as follows.

Proof. (1) The forward implication is an exercise. For the reverse, let $\Theta _{1}=\operatorname {\mathrm {IRS}}(\sigma _{\omega }), \Theta _{2}=\operatorname {\mathrm {IRS}}(\psi _{\omega })$ . For every finite $F\subseteq G$ , we have

$$ \begin{align*} \Theta_{1}(\{H:F\subseteq H\})=\int {\mathcal E}_{F}\,d\Theta_{1}&=\lim_{n\to\omega}\frac{1}{d_{n}}|\{j:\sigma_{n}(g)(j)=j \text{ for all } g\in F\}| \\ &=\lim_{n\to\omega}\frac{1}{d_{n}}|\{j:\psi_{n}(g)(j)=j \text{ for all } g\in F\}|\\ &=\int {\mathcal E}_{F}\,d\Theta_{2}=\Theta_{2}(\{H:F\subseteq H\}). \end{align*} $$

The $*$ -subalgebra $A_{0}=\operatorname {\mathrm {span}}\{\mathcal {E}_{F}:F\subseteq G\text { is finite}\}$ of $C(\operatorname {\mathrm {Sub}}(G))$ contains $1$ and separates points, and is thus dense by Stone and Weierstrass. It follows by the density and the Riesz representation theorem that $\Theta _{1}=\Theta _{2}$ . The result now follows by Theorem 5.1.

(2) The forward implication is an exercise. For the reverse, suppose that

$$ \begin{align*} \lim_{n\to\infty}\bigg|\frac{1}{d_{n}}\bigg|\{j:\sigma_{n}(g)(j)=j \text{ for all } g\in F\}\bigg| - \frac{1}{d_{n}}\bigg|\{j:\psi_{n}(g)(j)=j \text{ for all } g\in F\}\bigg|\bigg|=0 \end{align*} $$

for every finite $F\subseteq G$ . To show that $\psi _{n},\sigma _{n}$ are conjugate, it suffices by a diagonal argument to show that for every finite $E\subseteq G$ ,

$$ \begin{align*}\lim_{n\to\infty}\inf_{\chi\in {\operatorname{Sym}}(d_{n})}\sum_{g\in E}d_{\operatorname{\mathrm{Hamm}}}(\chi\sigma_{n}(g)\chi^{-1},\psi_{n}(g))=0.\end{align*} $$

If this does not hold, then there is an increasing sequence $n_{1}<n_{2}<\cdots $ and an $\varepsilon>0$ with

$$ \begin{align*}\inf_{\chi\in {\operatorname{Sym}}(d_{n})}\sum_{g\in E}d_{\operatorname{\mathrm{Hamm}}}(\chi\sigma_{n_{k}}(g)\chi^{-1},\psi_{n_{k}}(g))\geq \varepsilon.\end{align*} $$

Let $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ with $\{n_{k}:k\in {\mathbb N}\}\in \omega .$ By Corollary 5.2 and our hypothesis, we see $(\sigma _{\omega }),(\psi _{\omega })$ are conjugate. Then there is a $\chi =(\chi _{n})_{n\to \omega }\in \prod _{n\to \omega }({\operatorname {Sym}}(d_{n}),d_{\operatorname {\mathrm {Hamm}}})$ so that $\chi \sigma _{\omega }(g)\chi ^{-1}=\psi _{\omega }(g)$ for all $g\in G$ .

In particular,

$$ \begin{align*}L=\bigg\{n:\sum_{g\in E}d_{\operatorname{\mathrm{Hamm}}}(\chi_{n_{k}}\sigma_{n_{k}}(g)\chi_{n_{k}}^{-1},\psi_{n_{k}}(g))<\varepsilon\bigg\}\end{align*} $$

is in $\omega $ and by our choice of $n_{k},\omega $ , it follows that $L\in \omega ^{c}$ . This contradicts $\omega $ being a filter.

We remark that our proof of item (1) goes through the fact that for a group G, any $\Theta \in \operatorname {\mathrm {IRS}}(G)$ is uniquely determined by

$$ \begin{align*}(\Theta(\{H:F\subseteq H\}))_{F\subseteq G \text{ finite}},\end{align*} $$

thus the data of the IRS and those of the action trace as given in [Reference Arzhantseva and Păunescu5] are the same.

At this stage, we need the following two propositions for additional applications. Recall that if $\sigma _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ is a sofic approximation and there is a $\Theta \in \operatorname {\mathrm {Sub}}(G)$ with $\operatorname {\mathrm {IRS}}(\sigma _{\omega })=\Theta $ for all $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ , then we say that $(\sigma _{n})_{n}$ has stabilizer type $\Theta $ . Recall that we do not define stabilizer type if $\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ is different for different choices of $\omega $ .

We also need the following analogue of [Reference Arzhantseva and Păunescu4, Proposition 6.1], which is proved exactly as in [Reference Becker, Lubotzky and Thom7, Lemma 7.6] using Proposition 5.3.

We now have the tools to explain why Theorem 5.1 implies [Reference Becker, Lubotzky and Thom7, Theorem 1.3].

Proof. First suppose that for every $\Theta \in \operatorname {\mathrm {IRS}}(G)$ , there is a sequence of positive integers $(d_{n})_{n}$ and a sequence of homomorphisms $\kappa _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ with $({\operatorname {Stab}}_{\kappa _{n}})_{*}(u_{d_{n}})\to _{n\to \infty }\Theta $ in the weak $^{*}$ -topology. Let $\sigma _{n}\colon G\to {\operatorname {Sym}}(k_{n})$ be asymptotic homomorphisms, and fix $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ . Let $\Theta =\operatorname {\mathrm {IRS}}(\sigma _{\omega })$ . By assumption, we can find a sequence of homomorphisms $\kappa _{n}\colon G\to {\operatorname {Sym}}(d_{n})$ with $({\operatorname {Stab}}_{\kappa _{n}})_{*}(u_{d_{n}})\to _{n\to \infty }\Theta $ in the weak $^{*}$ -topology. By Proposition 5.4, we may assume that $d_{n}=k_{n}$ . Set $\kappa _{\omega }=(\kappa _{n})_{n\to \omega }$ . Then by Theorem 5.1, $\sigma _{\omega },\kappa _{\omega }$ are conjugate. Since $\omega $ was arbitrary, this implies that there is a sequence of permutations $\chi _{n}\in {\operatorname {Sym}}(d_{n})$ with

$$ \begin{align*} d_{\operatorname{\mathrm{Hamm}}}(\sigma_{n}(g),\chi_{n}\kappa_{n}(g)\chi_{n}^{-1})\to_{n\to\infty}0\quad \text{for all } g\in G.\end{align*} $$

Since $g\mapsto \chi _{n}\kappa _{n}(g)\chi _{n}^{-1}$ are homomorphisms, this proves that G is permutation stable.

Now suppose that G is permutation stable. Let $\Theta \in \operatorname {\mathrm {IRS}}(G)$ , and choose an action $G\,{\curvearrowright }\, (X,\nu )$ with ${\operatorname {Stab}}_{*}(\nu )=\Theta $ (e.g. the $\Theta $ -Bernoulli action of G with base $([0,1],m)$ ). Let ${\mathcal R}$ be the orbit equivalence relation of $G\,{\curvearrowright }\, (X,\nu )$ . Since G is amenable, [Reference Ornstein and Weiss40], [Reference Ornstein and Weiss41, II §3] (see also [Reference Connes, Feldman and Weiss15]) imply that ${\mathcal R}$ is hyperfinite and thus sofic (see e.g. [Reference Păunescu42, Proposition 3.4]). By Proposition 3.6, this implies that for every free ultrafilter $\omega $ , we may find a trace-preserving homomorphism

$$ \begin{align*}\pi\colon L({\mathcal R})\to {\mathcal M}:=\prod_{n\to\omega}(M_{d_{n}}({\mathbb C}),\operatorname{\mathrm{tr}})\end{align*} $$

such that $\pi (L^{\infty }(X))\subseteq L^{\infty }({\mathcal L})$ and $\pi ([{\mathcal R}])\subseteq {\mathcal S}$ . For $g\in G$ , let $p_{g}=1_{\operatorname {\mathrm {Fix}}(g)}\in L^{\infty }(X)$ and apply [Reference Anantharaman Delaroche and Popa3, Lemma 5.4.2(i)] to find $B_{g,n}\subseteq \{1,\ldots ,d_{n}\}$ with $\pi (p_{g})=(1_{B_{g,n}})_{n\to \omega }$ . For $g\in G$ , let $\pi (\unicode{x3bb} (g))=(\sigma _{n}(g))_{n\to \omega }$ . We first note the following.

Claim. $u_{d_{n}}(B_{g,n}\Delta \operatorname {\mathrm {Fix}}(\sigma _{n}(g)))\to _{n\to \omega }0$ for every $g\in G$ . To prove the claim, note that

$$ \begin{align*} \unicode{x3bb}(g)p_{g}=p_{g}\quad \text{and}\quad \tau(\unicode{x3bb}(g)(1-p_{g}))=0. \end{align*} $$

Let $p_{g,n}=1_{B_{g,n}}\in \ell ^{\infty }(d_{n})$ which we view as a subset of $M_{n}({\mathbb C}).$ Thus,

$$ \begin{align*}2u_{d_{n}}(\{j\in B_{g,n}:\sigma_{n}(g)(j)\ne j\})=\|\sigma_{n}(g)p_{g,n}-p_{g,n}\|_{2}^{2}\to_{n\to\omega}0,\end{align*} $$

the last part following as $\sigma _{n}$ is a sofic approximation and $\unicode{x3bb} (g)p_{g}=p_{g}$ . However,

$$ \begin{align*}u_{d_{n}}(\{j\in B_{g,n}^{c}:\sigma_{n}(g)(j)= j\})=\operatorname{\mathrm{tr}}(\sigma_{n}(g)(1-p_{g,n}))\to_{n\to\omega}\tau(\unicode{x3bb}(g)(1-p_{g}))=0,\end{align*} $$

the second-to-last part following as $\sigma _{n}$ is a sofic approximation. This proves the claim.

For $F\subseteq G$ , let $\mathcal {E}_{F}\in C(\operatorname {\mathrm {Sub}}(G))$ be the indicator function of $\{H:F\subseteq H\}$ . The claim similarly implies that for $F\subseteq G$ is finite if $\pi ({\mathcal E}_{F})=(1_{B_{F,n}})_{n\to \omega }$ , then

$$ \begin{align*}u_{d_{n}}\bigg(B_{F,n}\Delta\bigcap_{g\in F}\operatorname{\mathrm{Fix}}(\sigma_{n}(g))\bigg)\to_{n\to\omega}0.\end{align*} $$

Setting $\varsigma _{F,n}=\bigcap _{g\in F}\operatorname {\mathrm {Fix}}(\sigma _{n}(g))$ , we then have $\pi ({\mathcal E}_{F})=(1_{\varsigma _{F,n}})_{n\to \omega }$ . Since G is permutation stable, we may find honest homomorphisms $\kappa _{n}\colon G\to {\operatorname {Sym}}(n)$ so that $\sigma \circ \Xi =(\kappa _{n})_{n\to \omega }$ , where $\Xi \colon G\to [{\mathcal R}]$ is the map $\Xi (g)(x)=gx$ . So for any finite $F\subseteq G$ ,

$$ \begin{align*} \lim_{n\to\omega}({\operatorname{Stab}}_{\kappa_{n}})_{*}(\{H:F\subseteq H\})&=\lim_{n\to\omega}u_{d_{n}}(\{j:\kappa_{n}(g)(j)=j \text{ for all } g\in F\})\\ &=\lim_{n\to\omega}u_{d_{n}}(\{j:\sigma_{n}(g)(j)=j \text{ for all } g\in F\})\\ &=\lim_{n\to\omega}u_{d_{n}}(\varsigma_{F,n})\\ &=\lim_{n\to\omega}u_{d_{n}}(B_{F,n})\\ &=\tau(1_{{\mathcal E}_{F}})\\ &=\nu(\{x:gx=x \text{ for all } g\in F\})\\ &=\Theta(\{H:F\subseteq H\}). \end{align*} $$

As in Corollary 5.2, this implies that $\lim _{n\to \omega }({\operatorname {Stab}}_{\kappa _{n}})_{*}(u_{n})=\Theta .$ Since this holds for all $\omega $ , we may find a sequence of honest homomorphisms $\kappa _{n}\colon G\to {\operatorname {Sym}}(n)$ with

$$ \begin{align*} \lim_{n\to\infty}({\operatorname{Stab}}_{\kappa_{n}})_{*}(u_{n})=\Theta.\\[-40pt] \end{align*} $$

The proof of the reverse implication given above is similar to that in [Reference Arzhantseva and Păunescu5, Proposition 3.15].