Proof. Denote by $\ell _A$ the word length function on G associated to A.

Suppose first that G has property ${\mathscr S}$ -SC. Let $\varepsilon>0$ . Define $\Upsilon : {\mathcal F}(G)\rightarrow [0, \infty )$ by $\Upsilon (F)=\varepsilon ^{-1}|A|^{\max _{g\in F}\ell _A(g)}$ . Then there is an $S\in \overline {{\mathcal F}}(G)$ witnessing property ${\mathscr S}$ -SC. Let $T\in \overline {{\mathcal F}}(G)$ and $\delta>0$ . Then we have $C, n, S_1, \ldots , S_n$ as given by Definition 3.1. Set $m=\max _{1\le j\le n}\max _{g\in S_j}\ell _A(g)$ . Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation for G in ${\mathscr S}$ . Then we have W and ${\mathcal V}_1, \ldots , {\mathcal V}_n$ satisfying conditions (i)–(iii) in Definition 3.1. Denote by $V'$ the set of all $v\in V$ satisfying $\pi _{gh}v=\pi _g\pi _h v$ for all $g, h\in A^{mC}$ . When $\pi $ is a good enough sofic approximation, we have $\mathrm {m}(V\setminus V')\le \delta /|S|$ . Set $W'=W\cap V'$ . For each $1\le j\le n$ , set $m_j=\max _{g\in S_j}\ell _A(g)$ and ${\mathcal V}_j^\dagger =\bigcup _{g\in A^{m_j}}\pi _g{\mathcal V}_j$ . Set ${\mathcal V}=\bigcup _{j=1}^n{\mathcal V}_j^\dagger $ . Then

$$ \begin{align*} \mathrm{m}({\mathcal V})\le \sum_{j=1}^n\mathrm{m}({\mathcal V}_j^\dagger)\le \sum_{j=1}^n|A^{m_j}|\cdot \mathrm{m}({\mathcal V}_j)\le \varepsilon\sum_{j=1}^n\Upsilon(S_j)\mathrm{m}({\mathcal V}_j)\le \varepsilon, \end{align*} $$

verifying condition (i) in the proposition statement. Note also that

$$ \begin{align*}\mathrm{m}\bigg(\bigcup_{g\in S}\pi_gW'\bigg)\ge \mathrm{m}\bigg(\bigcup_{g\in S}\pi_g W\bigg)-|S|\cdot \mathrm{m}(V\setminus V')\ge 1-\delta,\end{align*} $$

which verifies condition (ii) in the proposition statement. Let $g\in T$ and $w_1, w_2\in W'$ be such that $\pi _g w_1=w_2$ . Then $w_1$ and $w_2$ are connected by a path of length at most C in which each edge is an $S_j$ -edge with both endpoints in ${\mathcal V}_j$ for some $1\le j\le n$ . It is easily checked that the endpoints of such an edge are connected by an A-path of length at most $m_j$ with all vertices in ${\mathcal V}_j^\dagger $ . Thus $w_1$ and $w_2$ are connected by an A-path of length at most $Cm$ with all vertices in ${\mathcal V}$ , verifying condition (iii) in the proposition statement. This proves the ‘only if’ part.

To prove the ‘if’ part, suppose that G satisfies the condition in the statement of the proposition. Let $\Upsilon $ be a function ${\mathcal F}(G)\rightarrow [0, \infty )$ . Take $0<\varepsilon <1/(2\Upsilon (A))$ . Then we have an S as in the statement of the proposition. Let $T\in \overline {{\mathcal F}}(G)$ . Take $0< \delta <1/(6|T|\Upsilon (T))$ . Then we have a C as in the statement of the proposition. Set $n=2$ , $S_1=A$ , and $S_2=T$ . Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation for G in ${\mathscr S}$ . Then we have W and ${\mathcal V}$ as in the statement of the proposition. Set $W'=W\cup \pi _{e_G}^{-1}(V\setminus \bigcup _{g\in S}\pi _g W)$ . Then $\bigcup _{g\in S}\pi _gW'=V$ , verifying condition (ii) in Definition 3.1. Set ${\mathcal V}_1={\mathcal V}$ and ${\mathcal V}_2=\bigcup _{g\in T}((W'\setminus W)\cup \pi _g(W'\setminus W)\cup \pi _g^{-1}(W'\setminus W))$ . Then

$$ \begin{align*} \mathrm{m}({\mathcal V}_2)\le (2|T|+1)\mathrm{m}(W'\setminus W)\le 3|T|\delta, \end{align*} $$

and hence

$$ \begin{align*} \Upsilon(S_1)\mathrm{m}({\mathcal V}_1)+\Upsilon(S_2)\mathrm{m}({\mathcal V}_2)\le \Upsilon(A)\varepsilon+3\Upsilon(T)|T|\delta< \tfrac12 + \tfrac12 = 1, \end{align*} $$

which verifies condition (i) in Definition 3.1. Let $g\in T$ and $w_1, w_2\in W'$ be such that $\pi _gw_1=w_2$ . If $w_1\not \in W$ or $w_2\not \in W$ , then $(w_1, w_2)$ is an $S_2$ -edge with both endpoints in ${\mathcal V}_2$ . If $w_1, w_2\in W$ , then $w_1$ and $w_2$ are connected by an $S_1$ -path of length at most C such that all vertices of this path lie in ${\mathcal V}={\mathcal V}_1$ , yielding condition (iii) in Definition 3.1.

Proof. Suppose to the contrary that G has property ${\mathscr S}$ -SC. Define $\Upsilon : {\mathcal F}(G)\rightarrow [0, \infty )$ by $\Upsilon (F)=4|G|$ for all $F\in {\mathcal F}(G)$ . Then there is some $S\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1. Put $T=\{e_G\}$ . Then there are $C, n\in {\mathbb N}$ and $S_1, \ldots , S_n\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1.

Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation in ${\mathscr S}$ so that there are subsets W and ${\mathcal V}_1, \ldots , {\mathcal V}_n$ of V satisfying conditions (i)–(iii) in Definition 3.1 and also so that $\mathrm {m}(U)\le 1/(4|G|)$ where U consists of all $v\in V$ satisfying $\pi _{e_G}v\neq v$ . Seeing that $W\subseteq U\cup \bigcup _{j=1}^n{\mathcal V}_j$ , we have

$$ \begin{align*} \mathrm{m}(W)&\le \mathrm{m}(U)+\mathrm{m}\bigg(\bigcup_{j=1}^n{\mathcal V}_j\bigg)\le \frac{1}{4|G|}+\sum_{j=1}^n\mathrm{m}({\mathcal V}_j)\\&=\frac{1}{4|G|}+\frac{1}{4|G|}\sum_{j=1}^n\Upsilon(S_j)\mathrm{m}({\mathcal V}_j)\le \frac{1}{2|G|}. \end{align*} $$

Thus

$$ \begin{align*} 1= \mathrm{m}\bigg(\bigcup_{g\in S}\pi_gW\bigg)\le |S|\mathrm{m}(W)\le |G|\mathrm{m}(W)\le \frac12 ,\end{align*} $$

a contradiction.

Proof. Suppose to the contrary that G has property ${\mathscr S}$ -SC. Then G must be infinite by Lemma 3.5. Take a strictly increasing sequence $\{G_k\}$ of finite subgroups of G such that $G=\bigcup _{k\in {\mathbb N}}G_k$ . For each $F\in {\mathcal F}(G)$ , denote by $\Phi (F)$ the smallest $k\in {\mathbb N}$ satisfying $F\subseteq G_k$ . Define $\Upsilon : {\mathcal F}(G)\rightarrow [0, \infty )$ by $\Upsilon (F)=3|G_{\Phi (F)}|$ . Then there is some $S\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1. Put $m=\Phi (S)$ and $T=G_{m+1}\in \overline {{\mathcal F}}(G)$ . Then there are $C, n\in {\mathbb N}$ and $S_1, \ldots , S_n\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1. Put $M=\max \{\max _{1\le j\le n}\Phi (S_j), m+1\}$ .

Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation in ${\mathscr S}$ so that there is a set $V_1\subseteq V$ satisfying the following conditions:

1. (i) $\pi _g\pi _hv=\pi _{gh}v$ for all $g, h\in G_M$ and $v\in V_1$ ;

2. (ii) $\pi _gv\neq \pi _hv$ for all $v\in V_1$ and distinct $g, h\in G_M$ ;

3. (iii) $\pi _gV_1=V_1$ for all $g\in G_M$ ;

4. (iv) $\mathrm {m}(V_1)\ge 1/2$ .

Then $G_M$ acts on $V_1$ via $\pi $ . Denote by ${\mathscr P}$ the partition of $V_1$ into $G_{m+1}$ -orbits.

By assumption, when $\pi $ is a good enough sofic approximation we can find subsets W and ${\mathcal V}_1, \ldots , {\mathcal V}_n$ of V satisfying conditions (i)–(iii) in Definition 3.1. Note that $V=\bigcup _{g\in S}\pi _gW=\bigcup _{g\in G_m}\pi _gW$ , which implies that for every member P of ${\mathscr P}$ the intersection $P\cap W$ is not contained in a single $G_m$ -orbit.

Set ${\mathcal V}=\bigcup _{j=1}^n\pi _{G_{\Phi (S_j)}}({\mathcal V}_j\cap V_1)$ . Then

(2) $$ \begin{align} \mathrm{m}({\mathcal V})\le \!\sum_{1\le j\le n}\kern-1pt|G_{\Phi(S_j)}|\mathrm{m}({\mathcal V}_j\cap V_1)\le \!\sum_{1\le j\le n}\kern-2.5pt|G_{\Phi(S_j)}|\mathrm{m}({\mathcal V}_j)= \frac{1}{3}\!\sum_{1\le j\le n}\kern-2pt\Upsilon(S_j)\mathrm{m}({\mathcal V}_j) \kern-1pt\le\kern-1pt \frac{1}{3}. \end{align} $$

Now let $P\in {\mathscr P}$ and $w_1\in P\cap W$ . Then we can find some $w_2\in P\cap W$ such that $w_1$ and $w_2$ are in different $G_m$ -orbits. We have $w_2=\pi _tw_1$ for some $t\in G_{m+1}\setminus G_m= T\setminus G_m$ . Thus we can find some $1\le l\le C$ , $1\le j_1, \ldots , j_l\le n$ , $v_k\in {\mathcal V}_{j_k}$ for $1\le k\le l$ , and $g_k\in S_{j_k}$ for $1\le k\le l$ such that, setting $v_0=w_1$ , we have $\pi _{g_k}v_{k-1}=v_k$ for all $1\le k\le l$ and $v_l=w_2$ . Then

$$ \begin{align*} w_2=v_l=\pi_{g_l}\cdots \pi_{g_1}v_0=\pi_{g_l\cdots g_1}w_1, \end{align*} $$

and hence $g_l\cdots g_1=t$ . It follows that the elements $g_1, \ldots , g_l$ cannot all lie in $G_m$ . Denote by i the smallest k satisfying $g_k\not \in G_m$ . Then $v_i\in \pi _{G_{\Phi (S_{j_i})}}w_1$ , and hence $w_1\in \pi _{G_{\Phi (S_{j_i})}}v_i$ . Consequently,

$$ \begin{align*} \pi_{G_m}w_1\subseteq \pi_{G_m}\pi_{G_{\Phi(S_{j_i})}}v_i=\pi_{G_{\Phi(S_{j_i})}}v_i\subseteq \pi_{G_{\Phi(S_{j_i})}}({\mathcal V}_{j_i}\cap V_1)\subseteq {\mathcal V}. \end{align*} $$

Therefore $V_1=\bigcup {\mathscr P}=\pi _{G_m}(W\cap (\bigcup {\mathscr P}))\subseteq {\mathcal V}$ , whence $ \mathrm {m}({\mathcal V})\ge \mathrm {m}(V_1)\ge 1/2$ , contradicting (2).

Proof. By Lemma 3.5 we may assume that G is infinite. Take a free subgroup $G_1$ of G with finite index. Then $G_1$ is non-trivial and, by Schreier’s lemma, finitely generated. Take free generators $a_1, \ldots , a_r$ for $G_1$ . Set $A=\{a_1, \ldots , a_r, a_1^{-1}, \ldots , a_r^{-1}, e_G\}$ . Denote by $\ell $ the word length function on $G_1$ associated to $a_1, \ldots , a_r, a_1^{-1}, \ldots , a_r^{-1}$ . For each $n\in {\mathbb N}$ denote by $B_n$ the set of elements g in $G_1$ satisfying $\ell (g)\le n$ . Take a subset H of G containing $e_G$ such that G is the disjoint union of the sets $hG_1$ for $h\in H$ . Set $D=H\cup A$ . For each $F\in {\mathcal F}(G)$ , denote by $\Psi (F)$ the smallest $n\in {\mathbb N}$ satisfying $F\subseteq HB_n$ , and set $F'=HB_{\Psi (F)}$ .

For any $g\in G$ and $h\in H$ we can write $gh$ uniquely as $bd$ with $b\in H$ and $d\in G_1$ , and using this factorization we set R to be the maximum value of $\ell (d)$ over all $g\in D$ and $h\in H$ . Define $\Upsilon : {\mathcal F}(G)\rightarrow [0, \infty )$ by $\Upsilon (F)=3|HB_{R}|\cdot |F'|$ .

Suppose that G has property ${\mathscr S}$ -SC. Then there is some $S\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1. Put $m=\Psi (S)$ , $N=m+1$ , and $T=S\{a_1^{2N}, e_G, a_1^{-2N}\}S\in \overline {{\mathcal F}}(G)$ . Then there are $C, n\in {\mathbb N}$ and $S_1, \ldots , S_n\in \overline {{\mathcal F}}(G)$ satisfying the conditions in Definition 3.1. Put $C'=C\max _{1\le j\le n}(1+\Psi (S_j))\in {\mathbb N}$ and $U=((HA)^{C'}HB_{2N})\cup ((HA)^{C'}HB_{2N})^{-1}\in \overline {{\mathcal F}}(G)$ .

Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation in ${\mathscr S}$ so that there are subsets W and ${\mathcal V}_1, \ldots , {\mathcal V}_n$ of V satisfying conditions (i)–(iii) in Definition 3.1 and a set $V'\subseteq V$ satisfying the following conditions:

1. (i) $\pi _g\pi _hv=\pi _{gh}v$ for all $g, h\in U^{10}$ and $v\in V'$ ;

2. (ii) $\pi _gv\neq \pi _hv$ for all $v\in V'$ and distinct $g, h\in U$ ;

3. (iii) $\mathrm {m}(V')\ge 1/2$ .

For each $1\le j\le n$ , set ${\mathcal V}_j'=\bigcup _{g\in S_j'}\pi _g{\mathcal V}_j.$ Set ${\mathcal V}=\bigcup _{j=1}^n{\mathcal V}_j'$ .

Let $v\in V'$ . We have $\pi _{a_1^{N}}v=\pi _{h_1}w_1$ and $\pi _{a_1^{-N}}v=\pi _{h_2}w_2$ for some $h_1, h_2\in S$ and $w_1, w_2\in W$ . Then

$$ \begin{align*} \pi_{h_2^{-1}a_1^{-2N}h_1}w_1=\pi_{h_2}^{-1}\pi_{a_1^{-N}}\pi_{a_1^{N}}^{-1}\pi_{h_1}w_1=\pi_{h_2}^{-1}\pi_{a_1^{-N}}v=w_2. \end{align*} $$

By assumption we can find a path from $w_1$ to $w_2$ of length at most C in which each edge is an $S_j$ -edge with both endpoints in ${\mathcal V}_j$ for some $1\le j\le n$ . Replacing each such edge by a D-path of length at most $1+\Psi (S_j)$ and with all vertices in ${\mathcal V}_j'$ , we find a D-path from $w_1$ to $w_2$ of length at most $C'$ such that all vertices are in ${\mathcal V}$ . Thus we get some $1\le l\le C'$ , $v_k\in {\mathcal V}$ for $1\le k\le l$ , and $g_k\in D$ for $1\le k\le l$ such that, setting $v_0=w_1$ , we have $\pi _{g_k}v_{k-1}=v_k$ for all $1\le k\le l$ and $v_l=w_2$ . Then

$$ \begin{align*} \pi_{h_2^{-1}a_1^{-N}}v=w_2=\pi_{g_l}\cdots \pi_{g_1}w_1=\pi_{g_l\cdots g_1h_1^{-1}a_1^{N}}v. \end{align*} $$

Since $h_2^{-1}a_1^{-N}$ and $g_l\cdots g_1h_1^{-1}a_1^{N}$ belong to U, we conclude that $h_2^{-1}a_1^{-N}=g_l\cdots g_1h_1^{-1}a_1^{N}$ . Set $t_j=g_j\cdots g_1h_1^{-1}a_1^{N}\in U$ for $0\le j\le l$ . We can write each $t_j$ uniquely as $b_jd_j$ for some $b_j\in H$ and $d_j\in G_1$ . Then we have

$$ \begin{align*} \ell(d_jd_{j-1}^{-1})\le R \end{align*} $$

for all $1\le j\le l$ . Consider the path p in $G_1$ from $d_0$ to $d_l$ defined by concatenating the geodesic from $d_{j-1}$ to $d_j$ for all $1\le j\le l$ , where we endow $G_1$ with the right invariant metric induced from $\ell $ . Note that as reduced words $d_0$ and $d_l$ end with $a_1$ and $a_1^{-1}$ , respectively. Thus p passes through $e_G$ . It follows that there is some $1\le i\le l$ with $\ell (d_{i})\le R$ . Then $t_i\in HB_{R}$ , whence

$$ \begin{align*} v=\pi_{t_i}^{-1}v_i=\pi_{t_i^{-1}}v_i\in \bigcup_{g\in (HB_{R})^{-1}}\pi_g {\mathcal V}. \end{align*} $$

Therefore $V'\subseteq \bigcup _{g\in (HB_{R})^{-1}}\pi _g {\mathcal V}$ .

Now we get

$$ \begin{align*} \frac12 \le \mathrm{m}(V') &\le \mathrm{m} \bigg(\bigcup_{g\in (HB_{R})^{-1}}\pi_g {\mathcal V} \bigg) \le |HB_{R}|\mathrm{m}({\mathcal V}) \\ &\le |HB_{R}|\sum_{j=1}^n|S_j'|\mathrm{m}({\mathcal V}_j) =\frac{1}{3}\sum_{j=1}^n\Upsilon(S_j)\mathrm{m}({\mathcal V}_j)\le \frac13 , \end{align*} $$

a contradiction.

Proof. We may assume, by passing to a suitable G-invariant conull subset of X, that the action of G is genuinely free. Let $\Upsilon $ be a function ${\mathcal F}(G)\rightarrow [0, \infty )$ . Since $G\curvearrowright (X, \mu )$ has property SC, using the function $2\Upsilon $ we find an $S\in \overline {{\mathcal F}}(G)$ such that for any $T\in \overline {{\mathcal F}}(G)$ there are $C, n\in {\mathbb N}$ , $S_1, \ldots , S_n\in \overline {{\mathcal F}}(G)$ , and Borel subsets W and ${\mathcal V}_k$ of X for $1\le k\le n$ satisfying the following conditions:

1. (i) $2\sum _{k=1}^n \Upsilon (S_k)\mu ({\mathcal V}_k)\le 1$ ;

2. (ii) $SW=X$ ;

3. (iii) if $w_1, w_2\in W$ satisfy $gw_1=w_2$ for some $g\in T$ then $w_1$ and $w_2$ are connected by a path of length at most C in which each edge is an $S_k$ -edge with both endpoints in ${\mathcal V}_k$ for some $1\le k\le n$ .

Let $T\in \overline {{\mathcal F}}(G)$ . Then we have $C, n$ , $S_k$ for $1\le k\le n$ , and W and ${\mathcal V}_k$ for $1\le k\le n$ as above. We now verify conditions (i)–(iii) in Definition 3.1 as referenced in Definition 3.3.

Let $g\in T$ . For each $x\in W\cap g^{-1}W$ , we can find $g_1, \ldots , g_l\in G$ for some $1\le l\le C$ such that $g=g_lg_{l-1}\cdots g_1$ and for each $1\le j\le l$ one has $g_j\in S_{k_j}$ and $g_{j-1}\cdots g_1x, g_jg_{j-1}\cdots g_1x\in {\mathcal V}_{k_j}$ for some $1\le k_j\le n$ . Then we can find a finite Borel partition ${\mathscr C}_g$ of $W\cap g^{-1}W$ such that

$$ \begin{align*} |{\mathscr C}_g|\le Cn^C\Big(\max_{1\le k\le n}|S_k|\Big)^C \end{align*} $$

and for each $A\in {\mathscr C}_g$ we can choose the same $l, g_1, \ldots , g_l, k_1, \ldots , k_l$ for all $x\in A$ . Then for all $1\le j\le l$ the sets $g_{j-1}\cdots g_1A$ and $g_jg_{j-1}\cdots g_1A$ are contained in ${\mathcal V}_{k_j}$ .

Denote by ${\mathscr C\,}^\sharp $ the finite partition of X generated by $W, {\mathcal V}_1, \ldots , {\mathcal V}_n$ and ${\mathscr C}_g$ for $g\in T$ . Set $F^\sharp =(T\cup S\cup \bigcup _{k=1}^nS_k)^{100C}\in {\mathcal F}(G)$ . Set $D=|T|C^2n^C(\max _{1\le k\le n}|S_k|)^C>0$ , and take $\delta>0$ with $3\delta |T|\Upsilon (T)\le 1/4$ . Take

$$ \begin{align*} 0<\delta^\sharp\le \min \bigg\{ \bigg(4\sum_{k=1}^n\Upsilon(S_k)\bigg)^{-1} , \delta/(|S|(|T|+D+2)) \bigg\}. \end{align*} $$

Let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a sofic approximation for G with $\operatorname {\mathrm {Hom}}_\mu ({\mathscr C\,}^\sharp , F^\sharp , \delta ^\sharp , \pi )\neq \emptyset $ which is good enough so that $\mathrm {m}(V_{F^\sharp })>1-\delta ^\sharp $ , where $V_{F^\sharp }$ denotes the set of all $v\in V$ satisfying $\pi _{gh}v=\pi _g\pi _hv$ for all $g, h\in F^\sharp $ and $\pi _gv\neq \pi _h v$ for all distinct $g, h\in F^\sharp $ . Take $\varphi \in \operatorname {\mathrm {Hom}}_\mu ({\mathscr C\,}^\sharp , F^\sharp , \delta ^\sharp , \pi )$ . Then $\varphi $ is an algebra homomorphism $\mathrm {alg}({\mathscr C\,}^\sharp _{F^\sharp })\rightarrow {\mathbb P}_V$ satisfying

1. (i) $\sum _{A\in {\mathscr C\,}^\sharp }\mathrm {m}(\pi _g\varphi (A)\Delta \varphi (gA))\le \delta ^\sharp $ for all $g\in F^\sharp $ , and

2. (ii) $\sum _{A\in {\mathscr C\,}^\sharp _{F^\sharp }}|\mathrm {m}(\varphi (A))-\mu (A)|\le \delta ^\sharp .$

Let $g\in T$ and $A\in {\mathscr C}_g$ . Then we have $l, g_1, \ldots , g_l, k_1, \ldots , k_l$ as above. Denote by $W_{g, A}''$ the set $\bigcup _{1\le j\le l}(\pi _{g_jg_{j-1}\cdots g_1}^{-1}(\varphi (g_jg_{j-1}\cdots g_1A))\Delta \varphi (A))$ . Then

$$ \begin{align*} \mathrm{m}(W_{g, A}'')&\le \sum_{j=1}^l\mathrm{m}(\pi_{g_jg_{j-1}\cdots g_1}^{-1}(\varphi(g_jg_{j-1}\cdots g_1A))\Delta \varphi(A))\\ &=\sum_{j=1}^l\mathrm{m}(\varphi(g_jg_{j-1}\cdots g_1A)\Delta \pi_{g_jg_{j-1}\cdots g_1}\varphi(A))\le C\delta^\sharp. \end{align*} $$

Also set $W_g''=\bigcup _{A\in {\mathscr C}_g}W_{g, A}''$ and $W''=\bigcup _{g\in T}W_g''$ . Then

$$ \begin{align*} \mathrm{m}(W_g'')\le |{\mathscr C}_g|C\delta^\sharp\le C^2n^C\Big(\max_{1\le k\le n}|S_k|\Big)^C\delta^\sharp \end{align*} $$

and

$$ \begin{align*} \mathrm{m}(W'')\le |T|C^2n^C\Big(\max_{1\le k\le n}|S_k|\Big)^C\delta^\sharp=D\delta^\sharp. \end{align*} $$

Set $W'=\bigcup _{g\in T}(\pi _g^{-1}(\varphi (W)\cap V_{F^\sharp })\setminus \varphi (g^{-1}W))$ . Note that

$$ \begin{align*} W'=\bigcup_{g\in T}(\pi_{g^{-1}}(\varphi(W)\cap V_{F^\sharp})\setminus \varphi(g^{-1}W))\subseteq \bigcup_{g\in T}(\pi_{g^{-1}}\varphi(W)\setminus \varphi(g^{-1}W)), \end{align*} $$

and hence

$$ \begin{align*} \mathrm{m}(W')\le \sum_{g\in T}\mathrm{m}(\pi_{g^{-1}}\varphi(W)\Delta \varphi(g^{-1}W))\le |T|\delta^\sharp. \end{align*} $$

Set $W^*=(\varphi (W)\cap V_{F^\sharp })\setminus (W'\cup W'')$ , and $W^\dagger =W^*\cup \pi _{e_G}^{-1}(V\setminus \bigcup _{g\in S}\pi _gW^*)$ . Then $\bigcup _{g\in S}\pi _gW^\dagger =V$ , verifying condition (ii) in Definition 3.1.

We have

$$ \begin{align*} \mathrm{m}\bigg(\bigcup_{g\in S}\pi_gW^*\bigg)&\ge \mathrm{m}\bigg(\bigcup_{g\in S}\pi_g\varphi(W)\bigg)-|S|\mathrm{m}(W'\cup W''\cup(V\setminus V_{F^\sharp}))\\ &\ge \mathrm{m}\bigg(\varphi\bigg(\bigcup_{g\in S}gW\bigg)\bigg)-\mathrm{m}\bigg(\bigg(\bigcup_{g\in S}\pi_g\varphi(W)\bigg)\Delta \varphi\bigg(\bigcup_{g\in S}gW\bigg)\bigg) \\ & \qquad\qquad\qquad\quad\quad \ -|S|(|T|\delta^\sharp+D\delta^\sharp+\delta^\sharp)\\ &= 1-\mathrm{m}\bigg(\bigg(\bigcup_{g\in S}\pi_g\varphi(W)\bigg)\Delta \bigcup_{g\in S}\varphi(gW)\bigg)-|S|(|T|+D+1)\delta^\sharp\\ &\ge 1-\sum_{g\in S}\mathrm{m}(\pi_g\varphi(W)\Delta \varphi(gW))-|S|(|T|+D+1)\delta^\sharp\\ &\ge 1-|S|\delta^\sharp-|S|(|T|+D+1)\delta^\sharp\ge 1-\delta, \end{align*} $$

and hence

$$ \begin{align*} \mathrm{m}(W^\dagger\setminus W^*)\le \mathrm{m}\bigg(V\setminus \bigcup_{g\in S}\pi_gW^*\bigg)\le \delta. \end{align*} $$

Put ${\mathcal V}_k^\dagger =\varphi ({\mathcal V}_k)$ for $1\le k\le n$ , $S_{n+1}=T\in \overline {{\mathcal F}}(G)$ , and

$$ \begin{align*} {\mathcal V}_{n+1}^\dagger=\bigcup_{g\in T}((W^\dagger\setminus W^*)\cup \pi_g(W^\dagger\setminus W^*)\cup \pi_g^{-1}(W^\dagger\setminus W^*)). \end{align*} $$

Then

$$ \begin{align*} \mathrm{m}({\mathcal V}_{n+1}^\dagger)\le (2|T|+1)\mathrm{m}(W^\dagger\setminus W^*)\le 3\delta |T|, \end{align*} $$

and hence

$$ \begin{align*} \sum_{k=1}^{n+1}\kern-1.5pt\Upsilon(S_k)\mathrm{m}({\mathcal V}_k^\dagger)\le 3\delta |T|\Upsilon(T)+\kern-1pt\sum_{k=1}^n\kern-1.5pt\Upsilon(S_k)(\mu({\mathcal V}_k)+\delta^\sharp)\le \frac14 + \frac12 +\delta^\sharp\kern-1pt\sum_{k=1}^n\kern-1.5pt\Upsilon(S_k)\le 1, \end{align*} $$

verifying condition (i) in Definition 3.1.

Let $g\in T$ and $w_1, w_2\in W^\dagger $ with $\pi _gw_1=w_2$ . If $w_1\not \in W^*$ or $w_2\not \in W^*$ , then $(w_1, w_2)$ is an $S_{n+1}$ -edge with both endpoints in ${\mathcal V}_{n+1}^\dagger $ . Thus we may assume that $w_1, w_2\in W^*$ . Then $w_1=\pi _g^{-1}w_2\in \pi _g^{-1}(\varphi (W)\cap V_{F^\sharp })$ . Since $w_1\notin W'$ , we get $w_1\in \varphi (g^{-1}W)$ . Thus

$$ \begin{align*} w_1\in \varphi(W)\cap \varphi(g^{-1}W)=\varphi(W\cap g^{-1}W)=\varphi\bigg(\bigcup_{A\in {\mathscr C}_g}A\bigg)=\bigcup_{A\in {\mathscr C}_g}\varphi(A). \end{align*} $$

We have $w_1\in \varphi (A)$ for some $A\in {\mathscr C}_g$ . Let $l, g_1, \ldots , g_l, k_1, \ldots , k_l$ be as above for this A. Then for all $1\le j\le l$ the sets $g_{j-1}\cdots g_1A$ and $g_jg_{j-1}\cdots g_1A$ are contained in ${\mathcal V}_{k_j}$ . Since $w_1\notin W_{g, A}''$ , we have $\pi _{g_jg_{j-1}\cdots g_1}w_1\in \varphi (g_jg_{j-1}\cdots g_1A)$ for all $1\le j\le l$ . Thus $\pi _{g_{j-1}\cdots g_1}w_1, \pi _{g_jg_{j-1}\cdots g_1}w_1\in \varphi ({\mathcal V}_{k_j})={\mathcal V}_{k_j}^\dagger $ for all $1\le j\le l$ . Therefore $w_1$ and $w_2$ are connected by a path of length l in which each edge is an $S_k$ -edge with both endpoints in ${\mathcal V}_k^\dagger $ for some $1\le k\le n$ , verifying condition (iii) in Definition 3.1.

Proof. The equivalence of (i)–(iii) is the content of [Reference Kerr and Li25, Proposition 3.5]. For (iii) $\Rightarrow $ (iv) apply Proposition 3.10. The implication (iv) $\Rightarrow $ (v) follows from the fact that non-trivial Bernoulli actions have positive sofic entropy with respect to every sofic approximation sequence [Reference Bowen5, Reference Kerr21, Reference Kerr and Li22], while (v) $\Rightarrow $ (iv) follows from the definitions and (v) $\Rightarrow $ (vi) from Propositions 3.6 and 3.7.

In the case that G is amenable, [Reference Kerr and Li25, Proposition 3.28] asserts that (vi) $\Leftrightarrow $ (i), which gives us the equivalence of all of the conditions.

Proof. Since G is not virtually cyclic, there exists a $c>0$ such that $|A^n| \geq cn^2$ for all $n\in {\mathbb N}$ [Reference Mann29, Corollary 3.5]. Set $b=5/c$ . Let $H, F, r, M, \delta $ , and $\pi $ be as in the lemma statement. Set $N=|A|^{3M}$ . Take $k\in {\mathbb N}$ such that $|A^{kr}|\ge N|A^{2r}|$ .

Denote by $V'$ the set of all $v\in V$ satisfying $\pi _{gh}v=\pi _g\pi _hv$ for all $g, h\in (F\cup A)^{100(M+r)}$ and $\pi _gv\neq \pi _hv$ for all distinct $g, h\in (F\cup A)^{100(M+r)}$ . Denote by $V''$ the set of all $v\in V$ satisfying $\pi _{gh}v=\pi _g\pi _hv$ for all $g, h\in (F\cup A)^{(200+k)(M+r)}$ and $\pi _gv\neq \pi _hv$ for all distinct $g, h\in (F\cup A)^{(200+k)(M+r)}$ . Then $\pi _gV''\subseteq V'$ for every $g\in A^{(k+6)r}$ . Assuming that $\pi $ is a good enough sofic approximation, we have $|V''|/|V|\ge 1-\delta $ . Take a maximal $(A,r)$ -separated subset W of $V''$ , and also take a maximal $(A, r)$ -separated subset $W'$ of $V'$ containing W. Then we have $\bigcup _{g\in A^{2r}}\pi _gW\supseteq V''$ , and hence

$$ \begin{align*} \frac{1}{|V|} \bigg|\bigcup_{g\in A^{2r}}\pi_gW\bigg| \ge \frac{|V''|}{|V|} \ge 1-\delta. \end{align*} $$

Let $w\in W$ . Set $T_w=W'\cap \pi _{A^{(k+2)r}}w$ . Note that $\pi _{A^{kr}}w\subseteq V'\subseteq \pi _{A^{2r}}W'$ . For each $g\in A^{kr}$ we have $\pi _gw\in \pi _{A^{2r}}z$ for some $z\in W'$ . Then $z\in \pi _{A^{2r}}\pi _gw\subseteq \pi _{A^{(k+2)r}}w$ , and hence $z\in T_w$ . Thus

$$ \begin{align*} \pi_{A^{kr}}w\subseteq \pi_{A^{2r}}T_w. \end{align*} $$

Therefore

$$ \begin{align*} |T_w|\ge \frac{|A^{kr}|}{|A^{2r}|}\ge N. \end{align*} $$

Set $T=\bigcup _{w\in W}T_w\subseteq W'$ . We have

$$ \begin{align*} |A^r| |W'| = \bigg| \bigsqcup_{w\in W'} \pi_{A^r} w \bigg| \leq |V| \end{align*} $$

whence

(3) $$ \begin{align} |T|\le |W'| \leq \frac{|V|}{|A^r|} \leq \frac{|V|}{cr^2}. \end{align} $$

Let $w\in W$ . Let $(T_w, E_w)$ be the graph whose edges are those pairs of vertices which can be joined by an A-path of length at most $4r+1$ , and let us show that it is connected. It is enough to demonstrate that a given $v\in T_w$ is connected to w by a path in $(T_w, E_w)$ . Choose a shortest A-path from w to v. For each vertex z in this path contained in $\pi _{A^{kr}}w$ , the fact that $z\in \pi _{A^{kr}}w\subseteq \pi _{A^{2r}}T_w$ means that we can connect z to some $u_z\in T_w$ by an A-path $p_z$ of length at most $2r$ . By inserting $p_z$ and its reverse at z, we construct an A-path from w to v in which points of $T_w$ appear in every interval of length $4r+1$ . Therefore v is connected to w by some path in $(T_w, E_w)$ , showing that $(T_w, E_w)$ is connected.

Consider the graph $(T , E )$ whose edges are those pairs of vertices which can be joined by an A-path of length at most $4r+1$ . From the above, every connected component of this graph has at least N points. Starting with $(T , E )$ , we recursively build a sequence of graphs with vertex set T by removing one edge at each stage so as to destroy some cycle at that stage, until there are no more cycles left and we arrive at a subgraph $(T, E')$ such that $(T, E)$ and $(T, E')$ have the same connected components and each connected component of $(T, E')$ is a tree.

For each pair $(v,w)$ in $E'$ , we choose an A-path in $\pi _{A^{4r}}T$ joining v to w of length at most $4r+1$ . Denote by ${\mathcal V}$ the collection of all vertices which appear in one of these paths. Then ${\mathcal V}\subseteq \pi _{A^{4r}}T\subseteq V'\subseteq V_F$ . Note that each A-connected component of ${\mathcal V}$ has at least N points, and $W\subseteq T\subseteq {\mathcal V}$ . Thus

$$ \begin{align*} \frac{1}{|V|} \bigg|\bigcup_{g\in A^{2r}}\pi_g{\mathcal V}\bigg| \ge \frac{1}{|V|} \bigg|\bigcup_{g\in A^{2r}}\pi_gW\bigg| \ge 1-\delta. \end{align*} $$

Moreover, using (3) we have

$$ \begin{align*} |{\mathcal V}| \leq |T|+4r|E'| \leq (4r+1)|T| \leq 5r\cdot \frac{|V|}{cr^2} =\frac{b|V|}{r}. \end{align*} $$

Let ${\mathcal C}$ be an A-connected component of ${\mathcal V}$ . Denote by $({\mathcal C}, E_{{\mathcal C}})$ the graph whose edges are the pairs $(w, v)\in {\mathcal C}^2$ such that $\pi _gw=v$ for some $g\in A$ . Then $({\mathcal C}, E_{{\mathcal C}})$ is connected. Endow ${\mathcal C}$ with the geodesic distance $\rho $ induced from $E_{{\mathcal C}}$ . Take a maximal subset $Z_{{\mathcal C}}$ of ${\mathcal C}$ which is $(\rho , M)$ -separated in the sense that the M-balls $\{ v\in {\mathcal C} : \rho (v,z) \leq M \}$ for $z\in Z_{{\mathcal C}}$ are pairwise disjoint. Then $Z_{{\mathcal C}}$ is $(\rho , 2M)$ -spanning in ${\mathcal C}$ , that is, every point of ${\mathcal C}$ is connected to some point of $Z_{\mathcal C}$ by an A-path of length at most $2M$ with all vertices in ${\mathcal C}$ . Since $|{\mathcal C}|\ge N=|A|^{3M}>|A|^{2M}$ , we have $|Z_{{\mathcal C}}|\ge 2$ . Then $|{\mathcal C}\cap \pi _{A^{M}}z|\ge M$ for every $z\in Z_{{\mathcal C}}$ . Since the sets ${\mathcal C}\cap \pi _{A^M}z$ for $z\in Z_{{\mathcal C}}$ are pairwise disjoint, we get

$$ \begin{align*} |Z_{{\mathcal C}}|M\le \sum_{z\in Z_{{\mathcal C}}}|{\mathcal C}\cap \pi_{A^M}z|\le |{\mathcal C}|. \end{align*} $$

Denote by Z the union of the sets $Z_{{\mathcal C}}$ where ${\mathcal C}$ runs over all A-connected components of ${\mathcal V}$ . Then every point of ${\mathcal V}$ is connected to some point of Z by an A-path of length at most $2M$ with all vertices in ${\mathcal V}$ , and $|Z|/|V|\le 1/M$ .

Proof. Suppose first that $G^\flat $ is locally virtually cyclic. Then $G^\flat $ is amenable and, by Lemma 3.5 and Theorem 3.11, neither locally finite nor virtually cyclic. It follows by Theorem 3.12 that G has property sofic SC, as desired.

Suppose now that $G^\flat $ is not locally virtually cyclic. In this case we will first set out the argument under the assumption that $G^\flat $ is normal in G. Take a finitely generated subgroup $G_0$ of $G^\flat $ such that $G_0$ is not virtually cyclic. Take an $S_1\in \overline {{\mathcal F}}(G_0)$ generating $G_0$ . Let $b> 0$ be as given by Lemma 3.13 for the group $G_0$ and generating set $S_1$ .

Let $\Upsilon $ be a function ${\mathcal F}(G)\rightarrow [0, \infty )$ . Choose an $r\in {\mathbb N}$ large enough so that

(4) $$ \begin{align} 3b\Upsilon(S_1)\leq r. \end{align} $$

Set $S=S_1^{2r}\in \overline {{\mathcal F}}(G_0)$ .

Consider the restriction of $3\Upsilon $ to ${\mathcal F}(G^\flat )$ . Since $G^\flat $ has property sofic SC, there exists an $S^\flat \in \overline {{\mathcal F}}(G^\flat )$ such that for any $T^\flat \in \overline {{\mathcal F}}(G^\flat )$ there are $C^\flat , n^\flat \in {\mathbb N}$ and $S^\flat _1, \ldots , S^\flat _{n^\flat }\in \overline {{\mathcal F}}(G^\flat )$ such that for any good enough sofic approximation $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ for G there are subsets $W^\flat $ and ${\mathcal V}^\flat _k$ of V for $1\le k\le n^\flat $ satisfying the following conditions:

1. (i) $\sum _{k=1}^{n^\flat } 3\Upsilon (S^\flat _{k})\mathrm {m}({\mathcal V}^\flat _{k})\le 1$ ;

2. (ii) $\bigcup _{g\in S^\flat }\pi _gW^\flat =V$ ;

3. (iii) if $w_1, w_2\in W^\flat $ satisfy $\pi _gw_1=w_2$ for some $g\in T^\flat $ then $w_1$ and $w_2$ are connected by a path of length at most $C^\flat $ in which each edge is an $S^\flat _k$ -edge with both endpoints in ${\mathcal V}^\flat _k$ for some $1\le k\le n^\flat $ .

Let $T\in \overline {{\mathcal F}}(G)$ . Set

$$ \begin{align*} S_2=S^\flat T S^\flat \in \overline{{\mathcal F}}(G). \end{align*} $$

Take an $M\in {\mathbb N}$ large enough so that

(5) $$ \begin{align} M\ge 12\Upsilon(S_2)|S_2|. \end{align} $$

Set $T^\flat =\bigcup _{g\in T}(S^\flat S_1^{2M}gS_1^{2M}g^{-1}S^\flat \cup S^\flat gS_1^{2M}g^{-1}S_1^{2M}S^\flat )\in \overline {{\mathcal F}}(G^\flat )$ . Then we have $C^\flat $ , $n^\flat $ , and $S^\flat _{k}$ for $1\le k\le n^\flat $ as above. Set

$$ \begin{align*} C=4M+2+C^\flat\in {\mathbb N}, \end{align*} $$

and $F=(S_1\cup T\cup S^\flat \cup \bigcup _{k=1}^{n^\flat }S_k^\flat )^{100MCr}\in {\mathcal F}(G)$ .

Now let $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ be a good enough sofic approximation for G. By Lemma 3.13 we can find sets $Z\subseteq {\mathcal V}_1\subseteq V_F$ , where $V_F$ denotes the set of $v\in V$ satisfying $\pi _{gh}=\pi _g\pi _hv$ for all $g, h\in F$ and $\pi _gv\neq \pi _hv$ for all distinct $g, h\in F$ , such that $\mathrm {m}(\bigcup _{g\in S}\pi _g{\mathcal V}_1)\ge 1-1/M$ , $\mathrm {m}({\mathcal V}_1)\le b/r$ , $\mathrm {m}(Z)\le 1/M$ , and every point of ${\mathcal V}_1$ is connected to some point of Z by an $S_1$ -path of length at most $2M$ with all vertices in ${\mathcal V}_1$ . Note that

Set $W={\mathcal V}_1\cup \pi _{e_G}^{-1}(V\setminus \bigcup _{g\in S}\pi _g {\mathcal V}_1)\subseteq V$ . Then $\bigcup _{g\in S}\pi _g W=V$ , which verifies condition (ii) in Definition 3.1.

Set ${\mathcal V}_2=(\bigcup _{g\in T}((W\setminus {\mathcal V}_1)\cup \pi _g (W\setminus {\mathcal V}_1)\cup \pi _g^{-1}(W\setminus {\mathcal V}_1)))\cup \bigcup _{g\in S_2}\pi _g Z\subseteq V$ . Then

$$ \begin{align*} \mathrm{m}({\mathcal V}_2)\le 3|T|\mathrm{m}(W\setminus {\mathcal V}_1)+|S_2|\mathrm{m}(Z)\le 3|S_2|\mathrm{m}\bigg(\!V\setminus \bigcup_{g\in S}\pi_g {\mathcal V}_1\!\bigg)+|S_2|/M\le 4|S_2|/M, \end{align*} $$

and hence

Assuming that $\pi $ is a good enough sofic approximation for G, we have $W^\flat $ and ${\mathcal V}^\flat _{k}$ for $1\le k\le n^\flat $ as above, in which case

$$ \begin{align*} \sum_{k=1}^{n^\flat}\Upsilon(S^\flat_{k})\mathrm{m}({\mathcal V}^\flat_{k})\le \frac{1}{3}. \end{align*} $$

Putting the above estimates together, we get

$$ \begin{align*} \Upsilon(S_1)\mathrm{m}({\mathcal V}_1)+\Upsilon(S_2)\mathrm{m}({\mathcal V}_2)+\sum_{k=1}^{n^\flat}\Upsilon(S^\flat_{k})\mathrm{m}({\mathcal V}^\flat_{k})\le 1, \end{align*} $$

which verifies condition (i) in Definition 3.1.

Let $g\in T$ and $w_1, w_2\in W$ be such that $\pi _gw_1=w_2$ . If either $w_1\in W\setminus {\mathcal V}_1$ or $w_2\in W\setminus {\mathcal V}_1$ , then $(w_1, w_2)$ is an $S_2$ -edge with both endpoints in ${\mathcal V}_2$ . Thus we may assume that $w_1, w_2\in {\mathcal V}_1$ . For $i=1, 2$ , we can connect $w_i$ to some $z_i\in Z$ by an $S_1$ -path of length at most $2M$ with all vertices in ${\mathcal V}_1$ . Then $w_i=\pi _{t_i}z_i$ for some $t_i\in S_1^{2M}$ . We have $\pi _gz_1=\pi _{a_1} u_1$ for some $u_1\in W^\flat $ and $a_1\in S^\flat $ , and $z_2= \pi _{a_2} u_2$ for some $u_2\in W^\flat $ and $a_2\in S^\flat $ . Note that $a_1^{-1}g$ and $a_2^{-1}$ are both in $S_2$ . Since $\pi _{a_1^{-1}g}z_1=u_1$ , the pair $(z_1, u_1)$ is an $S_2$ -edge with both endpoints in ${\mathcal V}_2$ . Also, since $\pi _{a_2^{-1}}z_2=u_2$ the pair $(w_2, u_2)$ is an $S_2$ -edge with both endpoints in ${\mathcal V}_2$ . Note that

$$ \begin{align*} \pi_{a_2^{-1}t_2^{-1}gt_1g^{-1}a_1} u_1&=\pi_{a_2^{-1}}\pi_{t_2^{-1}}\pi_{g}\pi_{t_1}\pi_{g^{-1}}\pi_{a_1} u_1\\ &=\pi_{a_2^{-1}}\pi_{t_2^{-1}}\pi_g\pi_{t_1}z_1\\ &=\pi_{a_2^{-1}}\pi_{t_2^{-1}}\pi_g w_1\\ &=\pi_{a_2^{-1}}\pi_{t_2^{-1}}w_2\\ &=\pi_{a_2^{-1}}z_2\\ &=u_2. \end{align*} $$

Since $a_2^{-1}t_2^{-1}gt_1g^{-1}a_1\in S^\flat S_1^{2M}gS_1^{2M}g^{-1}S^\flat \subseteq T^\flat $ , this means that $u_2\in \pi _{T^\flat }u_1$ . Then $u_1$ and $u_2$ are connected by a path of length at most $C^\flat $ in which each edge is an $S^\flat _k$ -edge with both endpoints in ${\mathcal V}^\flat _k$ for some $1\le k\le n^\flat $ . Therefore $w_1$ and $w_2$ are connected by a path of length at most $4M+2+C^\flat =C$ in which each edge is either an $S_j$ -edge with both endpoints in ${\mathcal V}_j$ for some $1\le j\le 2$ or an $S^\flat _k$ -edge with both endpoints in ${\mathcal V}^\flat _k$ for some $1\le k\le n^\flat $ , verifying condition (iii) in Definition 3.1.

Notice that the set S used in the above verification of property sofic SC for G is contained in $G_0$ , which can be any non-virtually-cyclic finitely generated subgroup of $G^\flat $ , and only depends on the restriction of $\Upsilon $ to $\overline {{\mathcal F}}(G_0)$ . This has the consequence that if $G_1 , G_2 , \ldots $ is a sequence of countable groups such that $G_n$ is a normal subgroup of $G_{n+1}$ for each n and $G_1$ is not locally virtually cyclic and has property sofic SC then the group $\bigcup _{n=1}^\infty G_n$ has property sofic SC. Indeed, we can fix a finitely generated subgroup $G_1'$ of $G_1$ which is not virtually cyclic and apply the above argument recursively, taking $G_0 = G_1'$ , $G^\flat = G_n$ , and $G=G_{n+1}$ at the nth stage to deduce that $G_{n+1}$ has property sofic SC, and if the function $\Upsilon $ is taken at each stage to be the restriction of a prescribed function ${\mathcal F} (\bigcup _{n=1}^\infty G_n ) \to [0,\infty )$ then we can use the same set S for all n, showing that $\bigcup _{n=1}^\infty G_n$ has property sofic SC. It follows by ordinal well-ordering that if $G^\flat $ is merely assumed to be w-normal in G then we can still conclude that G has property sofic SC.

Proof. If G and H are both locally finite then $G\times H$ is locally finite and hence does not have property ${\mathscr S}$ -SC by Proposition 3.6. Suppose then that at least one of G and H is not locally finite. Take two non-trivial Bernoulli actions $G\curvearrowright (X,\mu )$ and $H\curvearrowright (Y,\nu )$ . By [Reference Kerr and Li25, Proposition 3.32] the p.m.p. action $G\times H\curvearrowright (X \times Y ,\mu \times \nu )$ given by $(g,h)(x,y) = (gx,hy)$ for all $g\in G$ , $h\in H$ , $x\in X$ , and $y\in Y$ has property SC, and hence has property sofic SC by Proposition 3.10.

By [Reference Bowen5, Reference Kerr21], for every finite partition ${\mathscr C}$ of X, $F\in {\mathcal F} (G)$ containing $e_G$ , and $\delta> 0$ one has $\operatorname {\mathrm {Hom}}_\mu ({\mathscr C} , F,\delta ,\pi ) \neq \emptyset $ for every sufficiently good sofic approximation $\pi $ for G, and for every finite partition ${\mathscr D}$ of Y, $L\in {\mathcal F} (H)$ containing $e_H$ , and $\delta> 0$ one has $\operatorname {\mathrm {Hom}}_\nu ({\mathscr D} , L,\delta ,\sigma ) \neq \emptyset $ for every sufficiently good sofic approximation $\sigma $ for H. Given such sofic approximations $\pi : G\to \operatorname {\mathrm {Sym}} (V)$ and $\sigma : H\to \operatorname {\mathrm {Sym}} (W)$ and $\varphi \in \operatorname {\mathrm {Hom}}_\mu ({\mathscr C} , F,\delta ,\pi )$ and $\psi \in \operatorname {\mathrm {Hom}}_\nu ({\mathscr D} , L,\delta ,\sigma )$ , we have a homomorphism $\zeta : \mathrm {alg} ({\mathscr C}_F \times {\mathscr D}_L ) = \mathrm {alg} (({\mathscr C}\times {\mathscr D} )_{F\times L} )\to {\mathbb P}_{V\times W}$ determined by $\zeta (C\times D) = \varphi (C)\times \psi (D)$ for $C\in {\mathscr C}_F$ and $D\in {\mathscr D}_L$ , and one can readily verify that $\zeta $ belongs to $\operatorname {\mathrm {Hom}}_{\mu \times \nu } ({\mathscr C}\times {\mathscr D} , F\times L,2\delta ,\pi \times \sigma )$ , showing that this set of homomorphisms is non-empty. Since the algebra of subsets of $X\times Y$ generated by products of finite partitions is dense in the $\sigma $ -algebra with respect to the pseudometric $d(A,B) = (\mu \times \nu )(A\Delta B)$ , it follows by a simple approximation argument that for every finite partition ${\mathscr E}$ of $X\times Y$ , finite set $e_{G\times H} \in K\subseteq G\times H$ , and $\delta> 0$ one has $\operatorname {\mathrm {Hom}}_{\mu \times \nu } ({\mathscr E} , K,\delta ,\pi \times \sigma )\neq \emptyset $ for all good enough sofic approximations $\pi : G\to \operatorname {\mathrm {Sym}} (V)$ and $\sigma : H\to \operatorname {\mathrm {Sym}} (W)$ . Since the action $G\times H\curvearrowright (X \times Y ,\mu \times \nu )$ has property sofic SC, it follows that $G\times H$ has property ${\mathscr S}$ -SC.

Proof. Denote by $V_F$ the set of all $v\in V$ satisfying $\pi _g\pi _hv=\pi _{gh}v$ for all $g, h\in F$ and $\pi _gv\neq \pi _hv$ for all distinct $g, h\in F$ . Then

$$ \begin{align*} \mathrm{m} (V\setminus V_F)\le 2|F|^2\tau'\le \frac{\tau}{11}. \end{align*} $$

Denote by $V_\varphi $ the set of all $v\in V$ satisfying $d(\varphi (\pi _gv), g\varphi (v))\le \eta _{L^2}$ for all $g\in F$ . Then

$$ \begin{align*} \mathrm{m} (V\setminus V_\varphi )\le |F|\bigg(\frac{\tau'}{\eta_{L^2}}\bigg)^2\le \frac{\tau}{8}. \end{align*} $$

For all $s, t\in L^2$ and $v\in V_F\cap V_\varphi $ , since $\lambda (t, \varphi (v))\in F$ we have

$$ \begin{align*} d(\varphi(\pi_{\lambda(t, \varphi(v))}v), \lambda(t, \varphi(v))\varphi(v))\le \eta_{L^2} \end{align*} $$

and hence

$$ \begin{align*} \lambda(s, \varphi(\pi_{\lambda(t, \varphi(v))}v))=\lambda(s, \lambda(t, \varphi(v))\varphi(v))=\lambda(s, t\varphi(v)), \end{align*} $$

which yields

$$ \begin{align*} \sigma^{\prime}_s\sigma^{\prime}_tv=\pi_{\lambda(s, \varphi(\sigma^{\prime}_tv))}\sigma^{\prime}_tv &=\pi_{\lambda(s, \varphi(\pi_{\lambda(t, \varphi(v))}v))}\pi_{\lambda(t, \varphi(v))}v\\ &=\pi_{\lambda(s, t\varphi(v))}\pi_{\lambda(t, \varphi(v))}v\\ &=\pi_{\lambda(s, t\varphi(v))\lambda(t, \varphi(v))}v\\ &=\pi_{\lambda(st, \varphi(v))}v\\ &=\sigma^{\prime}_{st}v , \end{align*} $$

so that

(6) $$ \begin{align} \rho_{\mathrm{Hamm}}(\sigma^{\prime}_s\sigma^{\prime}_t, \sigma^{\prime}_{st})\le \mathrm{m} (V\setminus (V_F\cap V_\varphi))\le \frac{\tau}{11}+\frac{\tau}{8}=\frac{19\tau}{88}. \end{align} $$

Note that $\sigma ^{\prime }_{e_H}=\pi _{e_G}$ . For each $t\in H$ choose a $\sigma _t\in \operatorname {\mathrm {Sym}}(V)$ such that $\sigma _tv=\sigma ^{\prime }_tv$ for all $v\in V$ satisfying $\sigma ^{\prime }_{t^{-1}}\sigma ^{\prime }_tv=v$ . For each $t\in L^2$ , taking $s=t^{-1}$ in (6) we conclude that

$$ \begin{align*} \rho_{\mathrm{Hamm}}(\sigma_t, \sigma^{\prime}_t)&\le \rho_{\mathrm{Hamm}}(\sigma^{\prime}_{t^{-1}}\sigma^{\prime}_t, \mathrm{id})\\ &\le \rho_{\mathrm{Hamm}}(\sigma^{\prime}_{t^{-1}}\sigma^{\prime}_t, \sigma^{\prime}_{e_H})+\rho_{\mathrm{Hamm}}(\sigma^{\prime}_{e_H}, \mathrm{id})\\ &\le \frac{19\tau}{88}+\rho_{\mathrm{Hamm}}(\pi_{e_G}, \mathrm{id})\\ &\le \frac{19\tau}{88}+\tau'\\ &\le \frac{19\tau}{88}+\frac{\tau}{22}=\frac{23\tau}{88}, \end{align*} $$

which in particular shows that $\rho _{\mathrm {Hamm}}(\sigma _t, \sigma ^{\prime }_t)<\tau $ . For all $s,t \in L$ we then have

$$ \begin{align*} \rho_{\mathrm{Hamm}}(\sigma_s\sigma_t, \sigma_{st})&\le \rho_{\mathrm{Hamm}}(\sigma_s, \sigma^{\prime}_s)+\rho_{\mathrm{Hamm}}(\sigma_t, \sigma^{\prime}_t)+\rho_{\mathrm{Hamm}}(\sigma^{\prime}_s\sigma^{\prime}_t, \sigma^{\prime}_{st})\\&\quad+\rho_{\mathrm{Hamm}}(\sigma_{st}, \sigma^{\prime}_{st})\\ &\le \frac{23\tau}{88} + \frac{23\tau}{88}+\frac{19\tau}{88}+\frac{23\tau}{88}=\tau. \end{align*} $$

For all distinct $s,t\in L^2$ , since $\varphi (V)\subseteq X_0$ we have $\sigma ^{\prime }_sv\neq \sigma ^{\prime }_tv$ for all $v\in V_F$ and hence

$$ \begin{align*} \rho_{\mathrm{Hamm}}(\sigma_s, \sigma_t)&\ge \rho_{\mathrm{Hamm}}(\sigma^{\prime}_s, \sigma^{\prime}_t)-\rho_{\mathrm{Hamm}}(\sigma_s, \sigma^{\prime}_s)-\rho_{\mathrm{Hamm}}(\sigma_t, \sigma^{\prime}_t)\\ &\ge \mathrm{m} (V_F)-\frac{23\tau}{88}-\frac{23\tau}{88}\\ &\ge 1-\frac{\tau}{11}-\frac{23\tau}{44}> 1-\tau. \\ \end{align*} $$

Proof Proof of Proposition 3.16

Let $\Upsilon _H$ be a function ${\mathcal F}(H)\rightarrow [0, \infty )$ . Define the function $\Upsilon _G: {\mathcal F}(G)\rightarrow [0, \infty )$ by $\Upsilon _G(F)=2\Upsilon _H(\kappa (F, X))$ .

Since the action $G\curvearrowright X$ has property sofic SC, there exists an $S_G\in \overline {{\mathcal F}}(G)$ such that for any $T_G\in \overline {{\mathcal F}}(G)$ there are $C_G, n_G\in {\mathbb N}$ , $S_{G, 1}, \ldots , S_{G, n_G}\in \overline {{\mathcal F}}(G)$ , $L_G\in \overline {{\mathcal F}}(G)$ , and $0<\tau _G<1$ such that, for any $(L_G, \tau _G)$ -approximation $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ for G with $\operatorname {\mathrm {Map}}_d(L_G, \tau _G, \pi )\neq \emptyset $ , there are subsets $W_G$ and ${\mathcal V}_{G, j}$ of V for $1\le j\le n_G$ satisfying the following conditions:

1. (i) $\sum _{j=1}^{n_G} \Upsilon _G(S_{G, j})\mathrm {m}({\mathcal V}_{G, j})\le 1$ ;

2. (ii) $\bigcup _{g\in S_G}\pi _gW_G=V$ ;

3. (iii) if $w_1, w_2\in W_G$ satisfy $\pi _gw_1=w_2$ for some $g\in T_G$ then $w_1$ and $w_2$ are connected by a path of length at most $C_G$ in which each edge is an $S_{G, j}$ -edge with both endpoints in ${\mathcal V}_{G, j}$ for some $1\le j\le n_G$ .

Set $S_H=\kappa (S_G, X)\in \overline {{\mathcal F}}(H)$ .

Let $T_H\in \overline {{\mathcal F}}(H)$ . Set $T_G=\lambda (T_H, X)\in \overline {{\mathcal F}}(G)$ . Then we have $C_G$ , $n_G$ , $S_{G, j}$ for $1\le j\le n_G$ , $L_G$ , and $\tau _G$ as above. Set $C_H=C_G$ , $n_H=n_G+1$ , $S_{H, j}=\kappa (S_{G, j}, X)\in \overline {{\mathcal F}}(H)$ for $1\le j\le n_G$ , and $S_{H, n_H}=T_H\in \overline {{\mathcal F}}(H)$ . Take $0<\delta _H<1/(6\Upsilon _H(T_H)|T_H|)$ . Also, set $A=L_G\cup S_G\cup T_G\cup \bigcup _{j=1}^{n_G}S_{G, j}\in {\mathcal F}(G)$ , $L_H=\kappa (A^{2(100+C_G)}, X)\in {\mathcal F}(H)$ , and

$$ \begin{align*} \tilde{\tau}_G &= \min\{(\tau_G/2)^2, \delta_H/(4|S_G|\cdot|A|^{2(100+C_G)})\}>0, \\ \tau_H &= \min\{\eta_{A^{2(100+C_G)}}\tilde{\tau}_G^{1/2}/(8|L_H|)^{1/2}, \tilde{\tau}_G/(22|L_H|^2), \tau_G/(2|L_H|^{1/2})\}>0. \end{align*} $$

Let $\sigma\kern-1.5pt :\! H\kern-1pt\rightarrow\kern-1pt \operatorname {\mathrm {Sym}}(\kern-0.5pt V\kern-0.5pt)$ be an $(\kern-0.8 ptL_H,\kern-1pt \tau _H\kern-0.8pt)$ -approximation for H with $\operatorname {\mathrm {Map}}_d(\kern-0.8pt L_H,\kern-1pt \tau _H/2,\kern-1pt \sigma \kern-0.8pt)\kern-1.5pt \neq\kern-1.5pt \emptyset $ . Choose a $\varphi \in \operatorname {\mathrm {Map}}_d(L_H, \tau _H/2, \sigma )$ . Since $X_0$ is dense in X, by perturbing $\varphi $ if necessary we may assume that $\varphi \in \operatorname {\mathrm {Map}}_d(L_H, \tau _H, \sigma )$ and $\varphi (V)\subseteq X_0$ . Define $\pi ': G\rightarrow V^V$ by

$$ \begin{align*} \pi^{\prime}_gv=\sigma_{\kappa(g, \varphi(v))}v \end{align*} $$

for all $v\in V$ and $g\in G$ . By Lemma 3.17 there is an $(A^{100+C_G}, \tilde {\tau }_G)$ -approximation $\pi : G\rightarrow \operatorname {\mathrm {Sym}}(V)$ such that $\rho _{\mathrm {Hamm}}(\pi _g, \pi ^{\prime }_g)\le \tilde {\tau }_G$ for all $g\in A^{100+C_G}$ . For each $g\in L_G\subseteq A^{100+C_G}$ we have

$$ \begin{align*} d_2(g\varphi, \varphi\pi_g)&\le d_2(g\varphi, \varphi\pi^{\prime}_g)+d_2(\varphi\pi^{\prime}_g, \varphi\pi_g)\\ &\le \bigg(\frac{1}{|V|}\sum_{v\in V}d(\kappa(g, \varphi(v))\varphi(v), \varphi(\sigma_{\kappa(g, \varphi(v))}v))^2\bigg)^{1/2}+\tilde{\tau}_G^{1/2}\\ &\le \bigg(\frac{1}{|V|}\sum_{v\in V}\sum_{t\in \kappa(g, X)}d(t\varphi(v), \varphi(\sigma_tv))^2\bigg)^{1/2}+\frac{\tau_G}{2}\\ &\le (\tau_H^2|L_H|)^{1/2}+\frac{\tau_G}{2} \\ &\le \tau_G. \end{align*} $$

Thus $\varphi \in \operatorname {\mathrm {Map}}_d(L_G, \tau _G, \pi )$ . Then we have $W_G$ and ${\mathcal V}_{G, j}$ for $1\le j\le n_G$ as above.

We now verify conditions (i)–(iii) in Definition 3.1 as referenced in Definition 3.2. Denote by $V_1$ the set of all $v\in V$ satisfying $\pi _{g_1g_2}v=\pi _{g_1}\pi _{g_2}v$ for all $g_1, g_2\in A^{100+C_G}$ . Then $\mathrm {m}(V\setminus V_1)\le |A|^{2(100+C_G)}\tilde {\tau }_G$ . Also, denote by $V_2$ the set of $v\in V$ satisfying $\pi _g\pi _{g_1}v=\pi ^{\prime }_g\pi _{g_1}v$ for all $g\in A$ and $g_1\in A^{C_G}$ . Then $\mathrm {m}(V\setminus V_2)\le \tilde {\tau }_G|A|^{1+C_G}$ . Set $W_H'=W_G\cap V_1\cap V_2$ , and $W_H=W_H'\cup \sigma _{e_H}^{-1}(V\setminus \bigcup _{h\in S_H}\sigma _hW_H')$ . Then $\bigcup _{h\in S_H}\sigma _hW_H=V$ , verifying condition (ii) in Definition 3.1.

Note that

$$ \begin{align*} \mathrm{m}(W_H\setminus W_H')&\le 1-\mathrm{m}\bigg(\bigcup_{h\in S_H}\sigma_hW_H'\bigg)\\ &\le 1-\mathrm{m}\bigg(\bigcup_{g\in S_G}\pi^{\prime}_g W_H'\bigg)\\ &=1-\mathrm{m}\bigg(\bigcup_{g\in S_G}\pi_g W_H'\bigg)\\ &\le |S_G|(\mathrm{m}(V\setminus V_1)+\mathrm{m}(V\setminus V_2))\\ &\le |S_G|(\tilde{\tau}_G|A|^{2(100+C_G)}+ \tilde{\tau}_G|A|^{1+C_G})\\ &\le \delta_H. \end{align*} $$

Put ${\mathcal V}_{H, j}={\mathcal V}_{G, j}$ for all $1\le j\le n_G$ , and

$$ \begin{align*} {\mathcal V}_{H, n_H}=\bigcup_{h\in T_H}((W_H\setminus W_H')\cup \sigma_h(W_H\setminus W_H')\cup \sigma_h^{-1}(W_H\setminus W_H')). \end{align*} $$

Then

$$ \begin{align*} \mathrm{m} ({\mathcal V}_{H, n_H})\le (2|T_H|+1)\mathrm{m} (W_H\setminus W_H')\le 3|T_H|\delta_H, \end{align*} $$

and hence

$$ \begin{align*} \sum_{j=1}^{n_H} \Upsilon_H(S_{H, j})\mathrm{m}({\mathcal V}_{H, j})&=\Upsilon_H(T_H)\mathrm{m}({\mathcal V}_{H, n_H})+\sum_{j=1}^{n_G} \Upsilon_H(\kappa(S_{G, j}, X))\mathrm{m}({\mathcal V}_{G, j})\\ &=\Upsilon_H(T_H)\mathrm{m}({\mathcal V}_{H, n_H})+\frac{1}{2}\sum_{j=1}^{n_G} \Upsilon_G(S_{G, j})\mathrm{m}({\mathcal V}_{G, j})\\ &\le 3\Upsilon_H(T_H)|T_H|\delta_H+\tfrac12 \le 1, \end{align*} $$

verifying condition (i) in Definition 3.1. Let $h\in T_H$ and $w_1, w_2\in W_H$ with $\sigma _hw_1=w_2$ . If $w_1\not \in W_H'$ or $w_2\not \in W_H'$ , then $(w_1, w_2)$ is an $S_{H, n_H}$ -edge with both endpoints in ${\mathcal V}_{H, n_H}$ . We may thus assume that $w_1, w_2\in W_H'$ . Then

$$ \begin{align*} w_2=\sigma_hw_1=\pi^{\prime}_{\lambda(h, \varphi(w_1))}w_1=\pi_{\lambda(h, \varphi(w_1))}w_1\in \pi_{T_G}w_1 \end{align*} $$

and so $w_1$ and $w_2$ are connected by a path of length at most $C_G$ in which each edge is an $S_{G, j}$ -edge with both endpoints in ${\mathcal V}_{G, j}$ for some $1\le j\le n_G$ . It is easily checked that such an edge is also an $S_{H, j}$ -edge. This verifies condition (iii) in Definition 3.1.