The Wreath Product of Two Sofic Groups is Sofic

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by  Ben Hayes

THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC
BENHAYESANDANDREWSALE
Abstract. Given soficapproximations forcountable, discrete groups G,H, wecon-
structasoficapproximationfortheirwreathproductG≀H.
6
1
0
Sofic groups,introducedby Gromov[10]anddevelopedby Weiss [19], area largeclass
2
of groups which can be approximated, in some sense, by finite groups.
n
There are many examples, including all amenable groups, all residually finite groups,
a
J and all linear groups (by Malcev’s Theorem). However, because of the weakness of the
approximation by finite groups, few permanence properties of soficity are properly un-
3
1 derstood. Relatively straight-forwardexamples include closure under direct product and
increasing unions, and the soficity of residually sofic groups. More substantial results

] generally require some amenability assumption. For example, an amalgamated product
R
of two sofic groups is know to be sofic if the amalgamated subgroup is amenable (see
G
[9],[15],[4],[17]). This was extended to encompass the fundamental groups of all graphs
.
h of groups with sofic vertex groups and amenable edge groups [3]. In the same paper,
t it is shown that the graph product of sofic groups is sofic. Also, if H is sofic and is a
a
m coamenable subgroup of G, then G is sofic too [8].
We provea new permanence result, namely that soficityis closedunder taking wreath
[
products:

1
v Theorem 1. Let G,H be countable, discrete, sofic groups. Then G≀H is sofic.
6
8 We remarkthat ourresult is generalandrequiresno amenability or residualfiniteness
2 assumptions. The special case of Theorem 1 when G is abelian was proved by Paunescu
3 [15],whousedmethodsofanalysisandthe notionofsoficequivalencerelationsdeveloped
0 by Elek and Lippner [5]. While finishing this paper, we learnt that Holt and Rees have
.
1 dealt with the case when H is residually finite and G sofic [12]. We prove the general
0 result directly, by constructing a sofic approximation for the wreath product, giving a
6
proof that is constructive, quantitative (see Proposition 2.1), and entirely self-contained.
1
The notion of hyperlinearlity gives a class of groups defined in a similar vein to sofic
:
v groups, but where they are approximated instead by unitary groups. Our construction
i
X extends to the situation where G is hyperlinear and H sofic, showing that G ≀ H is
hyperlinear, see [11].
r
a Viatheirapproximationsbyfinitegroups,soficgroupshaveapplicationstoproblemsof
currentmathematicalinterestinawideareaoffields. Soficgroupsarerelevanttoergodic
theorybecausethey arethelargestclassofgroupsforwhichBernoullishiftsareclassified
by their base entropy (see [1],[13]) and for which Gottschalk’s surjuncitivity conjecture
holds (see e.g. [10],[13]). In the study of group rings they are useful because they are
the largest class of groups for which Kaplansky’s direct finiteness conjecture (see [6]) is
known. In the field of L2–invariants, they are the largest class of groups for which the
determinant conjecture is known (see [7]), which is necessary to define L2–torsion (see
[14] Conjecture 3.94). They are also the largest class of groups for which an analogue of
Lu¨ck approximation is known (see [18]). We refer the reader to [14] for applications of
L2–invariantsto geometryandgrouptheory. See also[16],[2]for asurveyofsofic groups.
Date:January14,2016.
1
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 2
Acknowledgments. The first named author would like to thank Jesse Peterson for
askinghim ifwreathproducts ofsofic groupsaresofic atthe NCGOASpring Institute in
2012 at Vanderbilt University.
1. Preliminaries
We begin with the necessary definitions, as well as a useful lemma to help us identify
sofic approximations in wreath products.
Definition 1. LetAbe a finite set. The normalized Hamming distance, denotedd ,
Hamm
on Sym(A) is defined by
1
d (σ,τ)= |{a∈A:σ(a)6=τ(a)}|.
Hamm
|A|
Definition 2. Let G be a countable discrete group, F a finite subset of G, and ε > 0.
Fix a finite set A and a function σ: G→Sym(A). We say that σ is (F,ε)–multiplicative
if
max d (σ(g)σ(h),σ(gh))<ε.
Hamm
g,h∈F
We say that σ is (F,ε)–free if
min d (σ(g),Id)>1−ε.
Hamm
g∈F\{1}
We say that σ is an (F,ε)–sofic approximation if it is (F,ε)–multiplicative, (F,ε)–free,
and furthermore σ(1) = Id. Lastly, we say that G is sofic if for every finite F ⊆ G and
ε>0, there is a finite set A and an (F,ε)–sofic approximation σ: G→Sym(A).
Our aim is to use sofic approximations for G and H and build a sofic approximation
for G≀H. First recall that the wreath product is defined as
G≀H = G⋊H
H
M
where the action of h∈H is given via α ∈Aut( G), defined by
h H
αh (gx)x∈H =(ghL−1x)x∈H.
(cid:16) (cid:17)
A homomorphism π: G≀H → K, for some group K, can be decomposed into a pair of
homomorphisms π : G → K, π : H → K which satisfy the following equivariance
1 H 2
condition:
L
π (h)π (g)=π (α (g))π (h), for all h∈H,g ∈ G.
2 1 1 h 2 H
The following lemma gives an “approximate analogue” to this situation.
L
Lemma 1.1. Let G,H be countable, discrete groups. For every finite set F ⊆ G≀H
0
there are finite sets E ⊆ G,E ⊆H such that the following holds: Let ε>0 and Ω
1 H 2
be a finite set. Suppose σ: G≀H →Sym(Ω) is a map such that
L
• the restriction of σ to G is (E ,ε/6)–multiplicative,
1
H
M
• the restriction of σ to H is (E ,ε/6)–multiplicative,
2
• max d σ(g,h),σ(g,1)σ(1,h) <ε/6,
Hamm
g∈E1,h∈E2
(cid:0) (cid:1)
• max d (σ(1,h)σ(g,1),σ(α (g),1)σ(1,h))<ε/6.
Hamm h
g∈E1,h∈E2
Then σ is (F ,ε)–multiplicative.
0
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 3
Proof. After makingthe rightdefinitions forE ,E ,weapplythe triangleinequalitysev-
1 2
eraltimestoobtaintheresult. Werequirethatif(g,h),(gˆ,hˆ)areinF ,theng,gˆ,α (gˆ)∈
0 h
E , and h,hˆ ∈E . This is true if we define
1 2
E = α (g):h∈π (F )∪{1},g∈π (F ) ,
1 h H 0 G 0
E =π (F ).
2 (cid:8)H 0 (cid:9)
We leave verification that this is sufficient to the reader. (cid:3)
To see howthis givesan“approximateanalogue”ofthe situationfor homomorphisms,
noticethattheabovelemmasaysthatanapproximatehomomorphismσ: G≀H →Sym(A)
can be thought of as a pair of approximate homomorphisms σ : G → Sym(A),
1 H
σ : H →Sym(A) so that
2
L
σ (h)σ (g)≈σ (α (g))σ (g)
2 1 1 h 2
for all g in a large enoughfinite subset of G and all h in a large enough finite subset
H
ofH.(Hereweuse≈toindicatethatbothmapsagreeonasufficientlylargesubsetofA).
L
ItisnothardtoseethattheabovelemmaisvalidwithG≀H replacedwithanysemidirect
product and Sym(A) replaced with any group equipped with a bi-invariant metric.
2. The sofic approximation for wreath products
To facilitate our proof, we need to introduce some notation. Let G,H be countable
discretegroupsandσ : G→Sym(A), σ : H →Sym(B)be twofunctions(notassumed
A B
to be homomorphisms). For h∈H,b ∈B, define
0
(h)
σ : G→Sym A
A,b0
!
B
M
by
σ(h) (g) (a ) =(a ) ,
A,b0 b b∈B b b∈B
where (cid:16) (cid:17)
σ (g)(a ), if b=bσ (h)b ,
A b B 0
a =
b
(ab, otherwise.
Now suppose that E ⊆H is finite, and take a subset
b
B ⊆{b∈B :σ (h )b6=σ (h )b for all h ,h ∈E with h 6=h .}.
0 B 1 B 2 1 2 1 2
Note that σ(h1)(g ) and σ(h2)(g ) commute if b∈B ,g ,g ∈G,h ,h ∈H and h 6=h .
A,b0 1 A,b0 2 0 1 2 1 2 1 2
Thus it makes sense to define, for b∈B ,
0
σ : G→Sym A
A,b
!
E B
M M
by
σ (g ) = σ(x)(g ).
A,b x x∈E A,b x
(cid:16) (cid:17) xY∈E
In our applications σ will be a sofic approximation, so we can take B to make up the
B 0
majority of B. Thus σ will be defined for “most” b ∈ B. We package all these maps
A,b
together as a single map
σ : G→Sym A ⊕B
A
! !
E B
M M
by c
σ (g)(a),b , if b∈B
A,b 0
σ (g)(a,b)=
A
((cid:0)(a,b), (cid:1) if b∈B\B0.
c
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 4
We extend σ to G by declaring that σ (g)=Id if g ∈ G, but g ∈/ G. Now
A H A H E
define
L L L
c c
σ : H →Sym A ⊕B
B
! !
B
M
by
c
σ (h)(a,b)= a,σ (h)b .
B B
Finally, define
(cid:0) (cid:1)
c
σ: G≀H →Sym A ⊕B
! !
B
M
by
b
σ(g,h)=σ (g)σ (h).
A B
Note that σ is determined by the choice of E,B and the two maps σ and σ .
0 A B
b c c
Proposition 2.1. Let F ⊆ G ≀ H be finite and ε > 0. Then there are finite sets
E ⊆ G,Eb ⊆ H and an ε′ > 0 so that if σ : G → Sym(A) and σ : H → Sym(B)
A B A B
are (E ,ε′),(E ,ε′)–sofic approximations for G,H respectively, then σ is an (F,ε)–sofic
A B
approximation.
b
The remainder of this section is dedicating to proving Proposition 2.1. We will see
below that it is possible to compute an explicit upper bound on ε′. It will depend only
on ε and the set F.
Weremarkthatσ(1,1)=Idbyconstruction. Weneedtoshowitis(F,ε)–multiplicative
and (F,ε)–free. First we explain how to define the sets E, E and E .
A B
Let F ⊆ G≀ H be finite and ε > 0. Define projections π : G≀ H → G and
b G H
π : G≀H → H by π (g,h) = g,π (g,h) = h. Let E ,E be as in Lemma 1.1 for the
H G H 1 2
finite set F =F ∪{1}∪F−1. As in the proof of Lemma 1.1, we have L
0
E = α (g):h∈π (F ),g ∈π (F ) ,
1 h H 0 G 0
E =π (F ).
2 (cid:8)H 0 (cid:9)
Recallthatforg =(g ) ∈⊕ Gthesupportofg,denotedSupp(g),isthesetofx∈H
x x∈H H
with g 6=1. We set
x
E =E ∪ hSupp(g),
2
g∈[E1
h∈E2
E = g ∈G:(g ) ∈E ,
A x x x∈H 1
(cid:8) EB =E−1E. (cid:9)
Then our collections of finite sets satisfy the following properties, each of which we need
later on:
E ⊇ {g :(g )∈E ,x∈E},
A x x 1
E ⊇ hSupp(g) for all h∈E ,g ∈E ,
2 1
E ⊇ E ,
2
E ⊇ E∪E−1∪E−1E.
B
Choose ε′ so that 0 < ε′ < ε . Let σ : G → Sym(A), σ : H → Sym(B) be
48|E|2 A B
(E ,ε′),(E ,ε′)–sofic approximations respectively. Set
A B
B ={b∈B :σ (h )b6=σ (h )b for all h ,h ∈E,h 6=h },
01 B 1 B 2 1 2 1 2
B ={b∈B :σ (h h )b=σ (h )σ (h )b for all h ,h ∈E},
02 B 1 2 B 1 B 2 1 2
B =B ∩B .
0 01 02
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 5
Since σ is a sofic approximation, we can intuitively think of B as making up most of
B 0
the set B. Indeed, Lemma 2.2 below confirms this. Use the sets E ⊂ H,B ⊆ B and
0
the maps σ ,σ to define the maps σ ,σ ,σ, as constructed at the startof this section.
A B A B
We claim that the map σ is an (F,ε)–sofic approximation. We first make the following
preliminary observation.
c c b
Lemma 2.2. Let κ>0.bIf ε′ < κ then |B |≥(1−κ)|B|.
4|E|2 0
Proof. Note that
B = {b∈B :σ (h )b6=σ (h )b},
01 B 1 B 2
h1,\h2∈E
h16=h2
B = {b∈B :σ (h )σ (h )b=σ (h h )b}.
02 B 1 B 2 B 1 2
h1,\h2∈E
h16=h2
So
|B\B | |{b∈B :σ (h )b6=σ (h )b}|
01 B 1 B 2
≤ 1− .
|B| |B|
h1X,h2∈E(cid:18) (cid:19)
h16=h2
We have for all h ,h ∈E with h 6=h :
1 2 1 2
|{b∈B :σ (h )b6=σ (h )b}|
B 1 B 2
= d (σ (h ),σ (h ))
Hamm B 1 B 2
|B|
= d (σ (h )−1σ (h ),Id).
Hamm B 2 B 1
Since d is a invariant under left multiplication, and E ⊇E∪E−1 we have that
Hamm B
d (σ (h )−1,σ (h−1))=d (Id,σ (h )σ (h−1))<ε′.
Hamm B 2 B 2 Hamm B 2 B 2
Inserting this into the above two inequalities and using that d is invariant under
Hamm
right multiplication we see that:
|{b∈B :σ (h )b6=σ (h )b}|
B 1 B 2
= d (σ (h ),σ (h ))
Hamm B 1 B 2
|B|
= d (σ (h )−1σ (h ),Id)
Hamm B 2 B 1
> d (σ (h−1)σ (h ),Id)−ε′
Hamm B 2 B 1
≥ d (σ (h−1h ),Id)−2ε′
Hamm B 2 1
> 1−3ε′,
where in the last two lines we again use that E ⊇E∪E−1∪E−1E. Thus
B
|B |
01 ≥(1−3|E|2ε′).
|B|
Similarly, (E ,ε′)–multiplicativity of σ gives
B B
|B |
02 ≥1− 1−d σ (h h ),σ (h )σ (h ) ≥1−|E|2ε′.
Hamm B 1 2 B 1 B 2
|B|
h1X,h2∈E(cid:16) (cid:0) (cid:1)(cid:17)
This proves the Lemma. (cid:3)
Fix κ>0 withκ< ε andchooseε′ >0as inLemma 2.2, soε′ < ε . Tocomplete
12 48|E|2
the proof of Proposition2.1, we break it up into two steps: we firstshow that this choice
of ε′ gives us that σ is (F,ε)–multiplicative, and then show it is (F,ε)–free.
Step 1. We show that σ is (F,ε)–multiplicative.
b
b
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 6
To prove Step 1, we apply Lemma 1.1, verifying below the four necessary conditions.
We first check that the restriction to ⊕ G is (E ,ε/6)–multiplicative. Let g,g′ ∈ E ,
H 1 1
then
d σ (gg′),σ (g)σ (g′)
Hamm A A A
(cid:0) 1 (cid:1) (a,b):b∈B0,σA,b(g)σA,b(g′)6=σA,b(gg′)
≤c|A||B||Bc|(cid:12)(cid:12)MBcA⊕(B\B0)(cid:12)(cid:12)+ (cid:12)(cid:12)(cid:8) |A||B||B| (cid:9)(cid:12)(cid:12)
1 (cid:12) (cid:12)
≤κ+ (cid:12) d (σ (g)(cid:12)σ (g′),σ (gg′)).
(cid:12) Hamm A,b (cid:12)A,b A,b
|B|
bX∈B0
Write g = (g ),g′ = (g′). Recall, σ (g) = σ(x)(g ). In the following, we use
x x A,b x∈E A,b x
that the different terms in the product commute, to be precise: for x 6=x ∈E we have
1 2
[σ(x1)(g ),σ(x2)(g′ )]=1. This gives Q
A,b x1 A,b x2
d (σ (g)σ (g′),σ (gg′))=d σ(x)(g )σ(x)(g′), σ(x)(g g′) .
Hamm A,b A,b A,b Hamm A,b x A,b x A,b x x
!
x∈E x∈E
Y Y
Next, using the bi-invariance of the Hamming distance, the triangle inequality, and the
(E ,ε′)–multiplicativity of σ to see that
A A
d (σ (g)σ (g′),σ (gg′)) ≤ d σ(x)(g )σ(x)(g′),σ(x)(g g′)
Hamm A,b A,b A,b Hamm A,b x A,b x A,b x x
xX∈E (cid:16) (cid:17)
= d (σ (g )σ (g′),σ (g g′))
Hamm A x A x A x x
x∈E
X
< |E|ε′.
Thus we get the required multiplicativity:
ε
d (σ (gg′),σ (g)σ (g′))<κ+|E|ε′ < .
Hamm A A A
6
The fact that the restrictionto H is (E ,ε/6)–multiplicativeis more straight-forward.
c c 2c
Indeed, for h,h′ ∈E we have
2
d (σ (hh′),σ (h)σ (h′))=d (σ (hh′),σ (h)σ (h′))<ε′,
Hamm B B B Hamm B B B
where we note that we can use the multiplicative property of σ since E ⊆E .
B 2 B
c c c
The thirdconditionofLemma1.1is automaticallysatisfiedby σ, byconstruction. We
finish this step by verifying the bound on the Hamming distance between σ(1,h)σ(g,1)
and σ(α (g),1)σ(1,h) for h∈E ,g ∈E . Indeed, for such g,h we have
h 2 1 b
b b
d (σ (h)σ (g),σ (α (g))σ (h))
Hamm B A A h B
b b
1
≤c|A||Bc||B|(cid:12) c A!⊕cB\(B0∩σB(h)−1(B0))(cid:12)
(cid:12) MB (cid:12)
(cid:12)1 (cid:12)
+ (cid:12) (a,b):b∈B ∩σ (h)−1B ,(cid:12)σ (g)a6=σ (α (g))a
|A|(cid:12)|B||B| 0 B 0(cid:12) A,b A,σB(h)b h
1 (cid:12)(cid:8) (cid:9)(cid:12)
≤2κ+ (cid:12)(a,b):b∈B ∩σ (h)−1B ,σ (g)a6=σ (α (g))a .(cid:12)
|A||B||B| 0 B 0 A,b A,σB(h)b h
(cid:12)(cid:8) (cid:9)(cid:12)
Since Supp(α (g)) = hSu(cid:12)pp(g), and E contains both Supp(g) and hSupp(g), it follo(cid:12)ws
h
that for every b∈B ∩σ (h)−1(B ) we have
0 B 0
σA,σB(h)b(αh(g))= σA(x,)σB(h)b(gh−1x)= σA(h,xσ)B(h)b(gx).
x∈hSupp(g) x∈Supp(g)
Y Y
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 7
Note that we have used that σ (1)=Id to restrict the number of terms in the product.
A
We use that for h ∈ E (and hence for h ∈ E ) we have that σ(hx) (g) = σ(x)(g).
2 A,σB(h)b A,b
Inserting this into the above equation we see that
σ (α (g))= σ(x)(g)=σ (g).
A,σB(h)b h A,b A,b
x∈SYupp(g)
Returning to the above inequality, we have shown that the remaining sets involved are
empty, implying that
ε
d (σ (h)σ (g),σ (α (g))σ (h))<2κ< .
Hamm B A A h B
6
The hypotheses of Lemma 1.1 have been checked, completing Step 1.
c c c c
Step 2. We show that σ is (F,ε)–free.
Suppose (g,h)∈F. If h6=1, then
b
(a,b):σ(g,h)(a,b)6=(a,b) ⊇ A ⊕ b∈B :σ (h)b6=b .
B
!
B
(cid:8) (cid:9) M (cid:8) (cid:9)
Thus b
d σ(g,h),Id ≥d σ (h),Id ≥1−ε′ >1−ε.
Hamm Hamm B
We may therefore assu(cid:0)me that h(cid:1)= 1. When(cid:0)b ∈ B0, th(cid:1)e permutation σ(g,1) will act by
(a,b)7→(σ (g)a,b), fobr all a∈⊕ A. We then see that
A,b B
b
(a,b):σ(g,1)(a,b)=(a,b) ⊆ A ⊕ B\B ∪ (a,b):b∈B ,σ (g)a=a .
0 0 A,b
! !
B
(cid:8) (cid:9) M (cid:0) (cid:1) (cid:8) (cid:9)
Assumebg = (g ) 6= 1 and consider the proportion of elements fixed by σ(g,1). By the
x
above we have
1 (a,b):σ(g,1)(a,b)=(a,b) ≤κ+ 1 a∈ BA:σAb,b(g)a=a .
|A||B||B| |B| |A||B|
(cid:12)(cid:8) (cid:9)(cid:12) bX∈B0 (cid:12)(cid:12)(cid:8) L (cid:9)(cid:12)(cid:12)
Sinceg 6=1(cid:12),wecanfibndx ∈Supp(g).Fo(cid:12)reveryb∈B ,wehavethatσ (h )b6=σ (h )b
0 0 B 1 B 2
for every h ,h ∈ Supp(g) with h 6= h . It follows that the (σ (x )b)–coordinate of
1 2 1 2 B 0
σ (g)a is σ (g )a , where a is the (σ (x )b)–coordinate of a. Thus
A,b A x0 β β B 0
a∈ A:σ (g)a=a ⊆ A ⊕ a∈A:σ (g )a=a .
( A,b )   A x0
MB B\{σMB(x0)b} (cid:8) (cid:9)
 
So
a∈ A:σ (g)a=a
B A,b ≤1−d (σ (g ),1).
|A||B| Hamm A x0
(cid:12)(cid:8) L (cid:9)(cid:12)
(cid:12) (cid:12)
Since g ∈E we can use the freeness of σ , yielding
x0 A A
1 |B |
|{(a,b):σ(g,1)(a,b)=(a,b)}|≤κ+ 0 (1−d (σ (g),1))≤κ+ε′.
|A||B||B| |B| Hamm A
Equivalently, b
d (σ(g,1),Id)≥1−κ−ε′ >1−ε.
Hamm
This verifies that σ is (F,ε)–free, and thus completes the proof of Proposition 2.1, and
hence of Theorem 1. b
b
THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 8
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VanderbiltUniversity, Nashville,TN37240
E-mail address: [email protected], [email protected]