METRIC APPROXIMATIONS OF WREATH PRODUCTS

arXiv:1608.02610v2 [] 15 Sep 2017

BEN HAYES AND ANDREW W. SALE

Abstract. Given the large class of groups already known to be sofic, there is seemingly
a shortfall in results concerning their permanence properties. We address this problem
for wreath products, and in particular investigate the behaviour of more general metric
approximations of groups under wreath products.
Our main result is the following. Suppose that H is a sofic group and G is a countable,
discrete group. If G is sofic, hyperlinear, weakly sofic, or linear sofic, then GoH is also sofic,
hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant
metric approximations, extending a general construction of metric approximations for GoH
that uses soficity of H.

1. Introduction
Sofic groups, introduced by Gromov [Gro99] and developed by Weiss [Wei00], are a large
class of groups that can be approximated, in some sense, by finite groups. We consider sofic
groups, as well as several other classes of groups which can be similarly defined by metric
approximations, namely weakly sofic groups (introduced by Glebsky and Rivera [GR08]),
linear sofic groups (introduced by Arzhantseva and Paunescu [APa17]) and hyperlinear groups
(implicitly defined by Connes and explicitly by Rădulescu [Con76, Ra08]).
Via their approximations, sofic, hyperlinear, linear sofic, and weakly sofic groups have
applications to a wide area of fields. For example, sofic groups are relevant to ergodic theory [Bow10, KL11], topological dynamics, in particular Gottschalk’s surjunctivity conjecture
[Gro99, KL11], group rings and Kaplansky’s direct finiteness conjecture [ES04] (also for linear
sofic groups [APa17, Prop 2.6]), and L2 –invariants [ES05, L0̈2]. Hyperlinear groups are of
interest in operator algebras, particularly the Connes embedding theorem [Con76], and in
group theory, particularly for the Kevare conjecture [Pes08, Cor 10.4]. We refer the reader to
[Pes08, CL15] for surveys on sofic and hyperlinear groups.
There are many examples of sofic groups, including all amenable groups, all residually
finite groups, 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
understood. Relatively straightforward examples 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 of two sofic
groups is know to be sofic if the amalgamated subgroup is amenable (see [ES11, Pa11, DKP14,
Pop14]). This was extended to encompass the fundamental groups of all graphs of groups
with sofic vertex groups and amenable edge groups [CHR14]. In the same paper, it is shown
that the graph product of sofic groups is sofic. Also, if H is a subgroup of G that is sofic and
coamenable, then G is sofic too [ES06].
Our first result is a new permanence property for soficity that concerns wreath
Lproducts.
Recall that the wreath product of two groups G and H is the semidirect product H G o H.
The first-named author gratefully acknowledges support by NSF Grant DMS-1600802.
1

2

BEN HAYES AND ANDREW W. SALE

Theorem 1. Let G, H be countable, discrete, sofic groups. Then G o H is sofic.
When G is abelian, Theorem 1 was proved by Paunescu [Pa11], who used methods of
analysis and the notion of sofic equivalence relations developed by Elek and Lippner [EL10].
By Elek and Szabo [ES06, Theorem 1] it follows that G o H is sofic if G is sofic and H is
amenable. From this, we may apply the work of Vershik and Gordon on local embeddability
into finite groups to see that G o H is sofic if G is sofic and H is locally embeddable into
amenable groups (this follows from the proof of [VG97, Proposition 3]). We refer the reader
to Holt and Rees [HR17] for other results on metric approximation of wreath products (e.g.
for commutator-contractive length functions).
We remark that, using the Magnus embedding (see [Sal15] for both the original and a
modern geometric definition), Theorem 1 implies the following (in fact it follows from the
weaker version of Paunescu, mentioned above [Pa11]).
Corollary 2. Let N be a normal subgroup of a finite rank free group F , and let N 0 be the
derived subgroup of N . If F/N is sofic, then F/N 0 is sofic.
Proof. The Magnus embedding is F/N 0 ,→ Zr o (F/N ). Since soficity passes to subgroups we
therefore get the corollary from Theorem 1, or [Pa11].

Sofic groups are also weakly sofic, linear sofic and hyperlinear, and we ask to what extent
these properties are preserved by wreath products. Weakly sofic groups are a class of groups
which can be approximated by finite groups in a weaker sense than sofic groups, namely one
is allowed to approximate G by any finite group with any bi-invariant metric, instead of just
permutation groups with the Hamming distance, as is the case for soficity (see Section 4
for precise definitions). Linear soficity and hyperlinearity are each classes of groups which
can be approximated by linear groups—to be linear sofic requires approximation by general
linear groups with respect to the rank metric, while hyperlinearity requires approximation by
unitary groups in the normalized Hilbert-Schmidt distance.
Our techniques proving Theorem 1 generalize to give the following, broader result.
Theorem 3. Let G, H be countable, discrete groups and assume that H is sofic. Then:
(i) If G is sofic, then so is G o H,
(ii) If G is hyperlinear, then so is G o H,
(iii) If G is linear sofic, then so is G o H,
(iv) If G is weakly sofic, then so is G o H.
The proof of Theorem 3 is constructive, and is almost entirely self-contained, the exceptions
being the use of equivalent definitions of soficity, hyperlinearity, and linear soficity, and a result
used for (iii) that concerns the behaviour of Jordan blocks under tensor products. The first
step in the proof of Theorem 3 is a general result on metric approximations of groups, the
proof of which is quantitative (see Proposition 3.1).
Part (iv) follows immediately from Proposition 3.1, while the other three parts require
extra constructions.
We remark that the arguments in the matricial cases (ii),(iii) are more delicate, as each
of these arguments require tensor products of operators. In (iii), for example, linear soficity
of G allows us to find almost homomorphisms θ : G → GLn (F), for some field F, so that
1
n Rank(θ(g) − Id) is bounded away from zero for g ∈ G \ {1}. However, this property is not
stable under taking tensor products: for example if n1 Rank(θ(g)−5 Id) and n1 Rank(θ(h)− 51 Id)
are both small for some g, h ∈ G, then n12 Rank(θ(g) ⊗ θ(h) − Id) will be small. Because of
this issue, we have to remark that linear soficity in fact implies that we can find an almost
homomorphism θ : G → GLn (F) so that inf λ∈F\{0} n1 Rank(θ(g) − λ Id) is bounded away from

METRIC APPROXIMATIONS OF WREATH PRODUCTS

3

zero for g ∈ G \ {1}. A similar issue occurs in the hyperlinear case, where we find an almost
homomorphism θ : G → U(n) so that θ(g) stays a bounded distance away from the scalar
matrices (a result of Radulescu [Ra08] enables us to do this for case (ii)). In each of these
cases, forcing the image of our group elements to be far away from the scalars is a property
that is stable under tensor products. This is a direct computation in the unitary case, whereas
the argument that this is true in the general linear case is more involved (see Proposition 4.7).
The structure of the paper is as follows. Section 2 contains the definition of C–approximable
groups, and the definition of sofic groups that we use. This section also looks at how we may
determine that a map from a wreath product to a group is almost multiplicative, and how we
endow our wreath products with suitable metrics. Once this is established, we give the initial
construction of the metric approximations of a wreath product in Section 3, before extending
this to each of the specific cases of Theorem 3 in Section 4.
Acknowledgments. We wish to thank the diligence and thorough work of the anonymous
referee. The first-named author would like to thank Jesse Peterson for asking him if wreath
products of sofic groups are sofic at the NCGOA Spring Institute in 2012 at Vanderbilt
University.
2. Preliminaries
We begin with the necessary definitions, as well as a useful lemma to help us identify metric
approximations in wreath products. We first establish some notation.
Notation 1. Throughout we will use 1 to denote the identity element of a group (we expect
the reader to be able to infer which group it comes from), except when we talk of the identity
matrix, when we use Id.
The metric approximations, to which we have referred, can be defined as an embedding
of a group into a metric ultraproduct of groups, each with a given bi-invariant metric. Such
an embedding gives rise to a sequence of maps to the groups in the ultraproduct. It is these
maps on which we focus our attention.
Before we define key properties of these maps, we remind the reader that a metric d on a
group H is said to be bi-invariant if d(axb, ayb) = d(x, y) for all a, b, x, y ∈ H. Throughout
the paper we will work with such metrics and their corresponding length functions. We recall
that a function ` : H → [0, ∞) is a length function on H if:
• `(h) = `(h−1 ) for all h ∈ H,
• `(gh) ≤ `(g) + `(h) for all g, h ∈ H.
We say that ` is conjugacy-invariant if also `(xgx−1 ) = `(g) for all x, g ∈ H.
A conjugacy-invariant length function ` on G defines a bi-invariant metric by d(x, y) =
`(y −1 x). Conversely if G has a bi-invariant metric d, then `(x) = d(x, 1) is a conjugacyinvariant length function.
Notation 2. When we switch between metrics and length functions we will pair them up
with equivalent decorations on the notation. For example, a metric d0 will correspond to a
length function `0 .
Definition 1. Let H be a group with a bi-invariant metric d. Fix a group G and a function
θ : G → H.
(a) Given F ⊆ G and ε > 0 we say that θ is (F, ε, d)–multiplicative if θ(1) = 1 and

max d θ(gh), θ(g)θ(h) < ε.
g,h∈F

4

BEN HAYES AND ANDREW W. SALE

(b) Given F ⊆ G and a function c : G \ {1} → (0, ∞) we say that θ is (F, c, d)–injective if for
all g ∈ F \ {1}

d θ(g), 1 ≥ c(g).
We remark that we will use the phrases almost multiplicative and almost injective to mean
(F, ε, d)–multiplicative and (F, c, d)–injective respectively when we do not wish to specify
F, ε, c and d.
Definition 2. Let C be a class of pairs (H, d), where H is a group and d a bi-invariant metric
on H (the same group may appear multiple times in C with different metrics). We say that a
group G is C–approximable if there is a function c : G \ {1} → (0, ∞) so that for every finite
F ⊆ G and ε > 0 there is a pair (H, d) ∈ C and an (F, ε, d)–multiplicative function θ : G → H
which is also (F, c, d)–injective.
A special example of C–approximable groups are sofic groups, where C consists of the finite
symmetric groups paired with the normalized Hamming distance (see [ES05]).
Definition 3. Let A be a finite set. The normalized Hamming distance, denoted dHamm , on
Sym(A) is defined by
1
|{a ∈ A : π(a) 6= τ (a)}| .
dHamm (π, τ ) =
|A|
The corresponding length function is denoted `Hamm .
For our purposes, we will use an alternative (but equivalent) definition of soficity (see
[ES05, Thm. 1]).
Definition 4. 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)–free if
min `Hamm (σ(g)) > 1 − ε.

g∈F \{1}

We say that σ is an (F, ε)–sofic approximation if it is (F, ε, dHamm )–multiplicative, and (F, ε)–
free. 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 start with approximations for G and H and use them to build approximations for the wreath product G o H. We will use permutational wreath products in our
approximations, and we recall here the definitions. Note that all our wreath products are of
the restricted variety, meaning we use direct sums rather than direct products.
Definition 5. Let X be a set on which H acts. The permutational wreath product is defined
as
M
G oX H =
GoH
X
L
where the action of h ∈ H is given via αh ∈ Aut ( X G), defined by a coordinate shift:


αh (gx )x∈X = (gh−1 x )x∈X .

The regular wreath product G o H is defined as above, taking X = H with H acting on itself
by left-multiplication.
A homomorphism ϕ : L
G o H → K, for some group K, can be decomposed into a pair
of homomorphisms ϕ1 :
H G → K, ϕ2 : H → K which satisfy the following equivariance
condition:
L
ϕ2 (h)ϕ1 (g) = ϕ1 (αh (g))ϕ2 (h), for all h ∈ H, g ∈ H G.

METRIC APPROXIMATIONS OF WREATH PRODUCTS

5

The following lemma gives an analogue to this for the case of almost multiplicative maps.
Lemma 2.1. Let G, H be countable, discrete groups, and let
projH : G o H → H and projG : G o H →

M

G

H

be the natural projection maps (note that the latter is not a homomorphism). For a finite
subset F0 ⊆ G o H define subsets

E1 = αh (g) : h ∈ projH (F0 ) ∪ {1}, g ∈ projG (F0 ) ,
E2 = projH (F0 ).
Let ε > 0 and K be a group with a bi-invariant metric d. Suppose Θ : G o H → K is a map
with Θ(1) = 1 such that
M
• the restriction of Θ to
G is (E1 , ε/6, d)–multiplicative,
H

• the restriction of Θ to H is (E2 , ε/6, d)–multiplicative,
•
•

max


d Θ(g, h), Θ(g, 1)Θ(1, h) < ε/6,

max

d (Θ(1, h)Θ(g, 1), Θ(αh (g), 1)Θ(1, h)) < ε/6.

g∈E1 ,h∈E2

g∈E1 ,h∈E2

Then Θ is (F0 , ε, d)–multiplicative.
Proof. Note that if (g, h), (g 0 , h0 ) are in F0 , then g, g 0 , αh (g 0 ) ∈ E1 , and h, h0 ∈ E2 . Applying
the triangle inequality gives the result. Verification of this is left to the reader.

In our construction we use maps to groups of the form L oB Sym(B), for some group L
endowed with a bi-invariant metric. To make sense of the notions of almost multiplicativity
and almost injectivity we need a bi-invariant metric on this wreath product. The following proposition explains how we do this, using the language of length functions. This was
described independently by Holt and Rees [HR17, §5].
Proposition 2.2. Let L be a group with a conjugacy-invariant length function ` and suppose
that `(g) ≤ 1 for all g ∈ G. For a finite set B, define `˜ on L oB Sym(B) by

1 X
`˜ (kb )b∈B , τ = `Hamm (τ ) +
`(kb ).
|B|
b∈B
τ (b)=b

Then `˜ is a conjugacy-invariant length function.
L
Proof. We first show that `˜ is conjugacy-invariant. Fix h = (hb ), k = (kb ) ∈
B L and
π, τ ∈ Sym(B). Then

(k, τ )−1 (h, π)(k, τ ) = ατ −1 (k −1 h)ατ −1 π (k), τ −1 πτ .
Using the conjugacy-invariance of `Hamm we have:

1
`˜ (k, τ )−1 (h, π)(k, τ ) = `Hamm (π) +
|B|

X
b∈B
τ −1 πτ (b)=b

−1
`(kτ−1
(b) hτ (b) kπ τ (b) ).

6

BEN HAYES AND ANDREW W. SALE

Note that if τ −1 πτ (b) = b, then τ (b) = π −1 τ (b). We can use this to rewrite the summation
term above, and then use the conjugacy invariance of ` to further simplify it:
1
|B|

X

`(kτ−1
(b) hτ (b) kτ (b) ) =

b∈B
πτ (b)=τ (b)

1
|B|

X

`(hτ (b) ) =

b∈B
πτ (b)=τ (b)

1 X
`(hb ).
|B|
b∈B
π(b)=b

Thus we see that

1 X
˜ π).
`˜ (k, τ )−1 (h, π)(k, τ ) = `Hamm (π) +
`(hb ) = `(h,
|B|
b∈B
π(b)=b


˜ π) is similar.
The proof that `˜ (k, π)−1 = `(k,
We now prove the triangle inequality. Take h, k, π, τ as above. Then

1
` (k, τ )(h, π) = `Hamm (τ π) +
|B|
= `Hamm (τ π) +

1
|B|

X

`(kb hτ −1 (b) )

b∈B
π(b)=τ −1 (b)

X

`(kb hπ(b) )

b∈B
π(b)=τ −1 (b)

Let B̂ = {b ∈ B : π(b) = τ −1 (b) 6= b}. Then, using the fact that ` is bounded by 1 we get


X
X


1 
|B̂| +
`(k
)
+
`(hb )
` (k, τ )(h, π) ≤ `Hamm (τ π) +
b
.
|B| 
b∈B
τ (b)=b

b∈B
π(b)=b

From the above we see that it is enough to show that
`Hamm (τ π) +

|B̂|
≤ `Hamm (τ ) + `Hamm (π).
|B|

Using the definition of the Hamming distance, we need
|{b : π(b) 6= τ −1 (b)}| + |B̂| ≤ |{b ∈ B : τ (b) 6= b}| + |{b ∈ B : π(b) 6= b}|.
Since B̂ ⊆ {b ∈ B : π(b) = b}, we get that the above is the same as:
|{b : π(b) 6= τ −1 (b)}| ≤ |{b ∈ B : τ −1 (b) 6= b}| + |{b ∈ B : π(b) 6= b, π(b) 6= τ −1 (b)}|,
which we can deduce from the inclusion
{b : π(b) 6= τ −1 (b)} ⊆ {b ∈ B : τ −1 (b) 6= b} ∪ {b ∈ B : π(b) 6= b, π(b) 6= τ −1 (b)}.
This completes the proof of the triangle inequality and thus of Proposition 2.2.



3. Construction of the Approximation
In the following we let G, H, K be groups, B a finite set, and we suppose that functions
θ : G → K and σ : H → Sym(B) (not necessarily homomorphisms) are given.

METRIC APPROXIMATIONS OF WREATH PRODUCTS

7

3.1. Some intuition. To give some idea of the intuition behind the construction that follows,
consider first how one can think of an element of a wreath product G o H. One may consider
(g, h) ∈ G o H, where g = (gx )x∈H , as a journey through H, starting at the identity, finishing
at h, and picking up elements of G at selected points of H (namely, pick up gx at x whenever
gx 6= 1).
If both G, H are sofic, we wish to construct a finite model for G o H using symmetric groups
(here K = Sym(A)). A sofic approximation, roughly speaking, gives us a finite set (A or B),
inside of which a significant part of the set behaves like a prescribed finite subset of G or H
respectively (see e.g. [Gro99, p. 157], [ES04, Prop 4.4]). We ultimately seek such a set for
G o H, and first we may try to combine A and B in a way which mimics the wreath product
of groups. However this approach leads to a problem.
The problem is that in the approximation of H using B there is no prescribed point in B
representing the identity. Thus the “journey” through H from the identity to h will translate
to a “journey” in B from β to b = σ(h)β, where the choice of β is arbitrary, and may be
allowed to vary.
L
It is for this reasonLthat, in the construction below, we use B K, rather than just K, in
the wreath product ( B K) oB Sym(B) that we map into. This could be interpreted as using
one copy of K for each choice of “identity vertex” in B.
3.2. The construction. The aim of this section is to define an approximation of G o H into
a wreath product that is, in some sense, smaller, or more controllable, than G o H. In Section
4 this approximation is then used to prove each part of Theorem 3, composing it with a
further approximation into the specific type of group for each property. The only exception
is part (iv), where weak soficity follows immediately from this construction and Proposition
3.1. Given a group K and a set B, we consider the wreath product
!
!
M
M M
K oB Sym(B) =
K o Sym(B),
B

B

B

L

L
so π ∈ Sym(B) acts on B ( B K) by απ where

L
απ (kb )b∈B = (kπ−1 (b) )b∈B , if kb ∈ B K for all b ∈ B.
L L
L
Note that if we identify k ∈ B ( B K) with an element (kb,β )b,β∈B in B⊕B K, then
απ ((kb,β )) = (kπ−1 (b),β ).
L L
Notation 3. When we encounter sets of the form B ( B A), we will use roman subscripts
to identify the outer index, and L
greek subscripts to identify the inner index. For example, for
a = (ab )b∈B , we have each ab ∈ B A, expressed as ab = (ab,β )β∈B .
Given the maps θ : G → K and σ : H → Sym(B) and a finite subset E of H we define
M
Θ:GoH →
K oB Sym(B)
B

by Θ(g, h) = (θB (g), σ(h)), where θB is a map we proceed to define below.
We use the finite subset E ⊂ H to define a subset BE of B, given as the intersection
B = B1 ∩ B2 , where
B1 = {b ∈ B : σ(h1 )−1 b 6= σ(h2 )−1 b for all h1 , h2 ∈ E, h1 6= h2 },
B2 = {b ∈ B : σ(h1 h2 )−1 b = σ(h2 )−1 σ(h1 )−1 b for all h1 , h2 ∈ E}.

8

BEN HAYES AND ANDREW W. SALE

L L
If we consider an element of B ( B K) as a function B × B → K, where we follow the
convention of Notation 3, then θB is the map defined by

(gh )h∈H



(b, β) 7→ θ(gh0 ), if b ∈ BE , if gh = 1 for all h ∈ H \ E,


and h0 ∈ H is so that b = σ(h0 )β;
7→



(b, β) 7→ 1,
otherwise.

Referring back to the intuition of Section 3.1, we explain what happens whenL
we fix
Lβ, the
second coordinate in B ×B. This coordinate is the inner index for an element
of
(
B
B K),
L
and T
corresponds to a choice of a copy of K in the wreath product
K
o
Sym(B).
B
B
L If
β ∈ x∈E σ(x)−1 BE , then θB restricts to a map that sends (gx )x∈E to an element of B K,
where for each x ∈ E, the σ(x)β–coordinate is given by θ(gx ), and coordinates not of the
form σ(x)β for x ∈ E are trivial. Thus, we can think of β as behaving as the chosen “identity
vertex” in B. If an element (g, h) of G o H is a journey through H, starting at 1 and picking
up elements of G en route to h, then under Θ this turns into a collection of journeys through
B, each starting at a suitable choice of β, and finishing at σ(h)β. The map θB tells you what
elements of K to pick up along the way. If the original journey visited a vertex x ∈ E, then
the image journey starting at β will pick up θ(gx ) at σ(x)β. This is visualized in Figure 1

H

h

σ(h)β

E

B
β

1

x

σ(x)β

Figure
L 1. The journey through H corresponding to (g, h) on the left, when
g ∈ E H, and the journey in the image of Θ corresponding to choosing β to
play the role of the identity. The image Θ(g, h) will be made up of multiple
such journeys, one for each suitable choice of β.

METRIC APPROXIMATIONS OF WREATH PRODUCTS

9

We now give an equivalent definition of θB . This is necessary in order to establish further
maps and notation which will be used later on. For h ∈ H, b ∈ B, define
M
(h)
θb : G →
K
B

by

(h)
θb (g)

= (kβ )β∈B where
(
θ(g),
kβ =
1,
(h )

if β = σ(h)−1 b,
otherwise.

(h )

Note that θb 1 (g1 ) and θb 2 (g2 ) commute if b ∈ B1 , h1 , h2 ∈ E, g1 , g2 ∈ G and h1 6= h2 .
Thus it makes sense to define, for b ∈ BE ,
M
M
θb :
G→
K
E

B

by


Y (h)
θb (gh )h∈E =
θb (gh ).
h∈E

In our applications σ will be a sofic approximation, so we can think of BE as making up the
majority of B. Thus θb will be defined for “most” b ∈ B. We extend θb to be defined for all
b ∈ B by saying that θb maps everything to the identity for
L b ∈ B \ BE .
Relating this to the intuition described above, for g ∈ E G and β ∈ B, the β–coordinate
of θb (g) tells you what element of K to pick up at b L
if β is chosen
L asLthe “identity vertex.”
We then obtain our equivalent definition of θB :
EG →
B(
B K) by packaging all
these maps together as a single map

θB (g) = θb (g) b∈B
L
L
L
and extending θB to H G by declaring that θB (g) = 1 if g ∈ H G, but g ∈
/ E G.
We will prove that if K has a bi-invariant metric and θ and σ are almost multiplicative and
almost injective, then Θ gives us our desired almost multiplicative and almost
L injective map.
To do this, we need to use an appropriate bi-invariant length function on ( B K) oB Sym(B).
L
Notation 4. Let `0 be a conjugacy-invariant length function on B K such that `0 ≤ 1. Take
L
`˜ to be the conjugacy-invariant length function on ( B K) oB
LSym(B) defined in Proposition
2.2. When `0 is the specific length function `max defined on B K by

`max (kb )b∈B = max `(kb ),
b∈B

where ` is a given length function on K such that ` ≤ 1, we denote the length function
obtained from Proposition 2.2 by `˜max , to emphasise the specific choice of `0 .
In summary, we will be dealing with the following length functions, with corresponding
metrics:
• ` on K,
L corresponding to d, and such that ` ≤ 1;
• `0 on BLK, corresponding to d0 , and such that `0 ≤ 1;
• `max on B K—we never refer to the corresponding metric;
L
˜
• `˜ on ( B K) oB Sym(B), corresponding to d;
L
˜
• `max on (
K) oB Sym(B), corresponding to d˜max .
B

Our aim is to prove the following.

10

BEN HAYES AND ANDREW W. SALE

Proposition 3.1. Let F ⊆ G o H be finite and ε > 0. There are finite subsets EG ⊆ G and
E, EH ⊆ H, and an ε0 > 0 with the following properties. Let
• σ : H → Sym(B) be an (EH , ε0 )–sofic approximation,
• θ : G → K be a map,
˜ d,
˜ `˜max , and d˜max be as described in Notation 4.
• `, d, `0 , d0 , `,
L
Then Θ : G o H → ( B K) oB Sym(B), as constructed above using E, θ and σ, has the
following properties.
L
(a) Suppose the length function `0 on B K restricts to ` on each copy of K.
˜
If θ : G → K is (EG , ε0 , d)–multiplicative, then Θ is (F, ε, d)–multiplicative.
0
(b) Let c be a map c : G \ {1} → (0, ∞). Define c : (G o H) \ {1} → (0, ∞) by

1,
if h 6= 1
2
c0 (g, h) =
1
 max 2 c(gx ), if h = 1, g = (gx )x∈H .
x∈Supp(g)

Then, if θ : G → K is (EG , c, d)–injective then Θ is (F, c0 , d˜max )–injective.
The remainder of this section is dedicating to proving Proposition 3.1. We will see below
that, once E is given, the following upper bounds on ε0 are sufficient:
(1)

ε0 <
ε0 <

for (a):
for (b):

ε
48|E|2 ,

1
min c(g), 1
16|E|2

| g ∈ EG \ {1} .

As we see, the bounds on ε0 depend only on ε and the set F .
We remark that Θ(1, 1) = 1 by construction. We first explain how to define the sets E,
EG and EH .
L
Let F ⊆ G o H be finite and ε > 0. Define projections projG : G o H →
H G and
projH : G o H → H by projG (g, h) = g and projH (g, h) = h. Let E1 , E2 be as in Lemma 2.1
for the finite set F0 = F ∪ {1} ∪ F −1 :
M

E1 = αh (g) : h ∈ projH (F0 ), g ∈ projG (F0 ) ⊆
G, E2 = projH (F0 ) ⊆ H.
E

L

Recall that for g = (gx )x∈H ∈ H G the support of g, denoted Supp(g), is the set of x ∈ H
with gx 6= 1. We set
[

E = E2 ∪
h Supp(g), EG = gx ∈ G : (gx ) ∈ E1 , x ∈ H , EH = E −1 E.
g∈E1
h∈E2

Since E2 contains the identity, it follows that E and E −1 are both subsets of EH .
Let K be as in Proposition 3.1 and let θ : G → K be (EG , ε0 , d)–multiplicative and
(EG , c, d)–injective, where ε0 is controlled by the bounds in (1) above. Let σ : H → Sym(B) be
a (EH , ε0 )–sofic approximation. Recall the set BE is defined from E as the intersections of sets
B1 , B2 (which depend only on E). Lemma 3.2 confirms that, since σ is a sofic approximation,
BE makes up a significant proportion of the set B.
Lemma 3.2. Let κ > 0. If ε0 <

κ
4|E|2

then |B \ BE | ≤ κ|B|.

Proof. Note that
B \ B1 =

[
h1 ,h2 ∈E
h1 6=h2

{b ∈ B : σ(h1 )−1 b = σ(h2 )−1 b}.

METRIC APPROXIMATIONS OF WREATH PRODUCTS

11

0
Since EH ⊇ E ∪ E −1 , by (EH , ε0 )–soficity of σ we have dHamm (σ(h2 )−1 , σ(h−1
2 )) < ε . Thus
for h1 6= h2 , we have

{b ∈ B : σ(h1 )−1 b = σ(h2 )−1 b}
|B|

=

dHamm (σ(h1 )−1 , σ(h2 )−1 )

=

1 − `Hamm (σ(h1 )σ(h2 )−1 )

<

0
1 − `Hamm (σ(h1 )σ(h−1
2 )) + ε

≤

0
1 − `Hamm (σ(h1 h−1
2 )) + 2ε

<

3ε0 ,

where in the last two lines we again use that EH ⊇ E ∪ E −1 ∪ E −1 E. Thus
|B \ B1 |
2
≤ 3 |E| ε0 .
|B|
Similarly, (EH , ε0 , dHamm )–multiplicativity of σ gives
X 

|B \ B2 |
2
≤
1 − dHamm σ(h1 h2 ), σ(h1 )σ(h2 ) ≤ |E| ε0 .
|B|
h1 ,h2 ∈E

This proves the lemma.



Use the set E ⊂ H and the maps θ, σ to define the maps θB , Θ, as constructed at the start
of this section.
3.3. Part (a) of Proposition 3.1. We claim that if ε0 is sufficiently small, then the map Θ
˜
is (F, ε, d)–multiplicative.
ε
Take κ > 0 so that κ < 12
, and take ε0 > 0 satisfying the hypothesis of Lemma 3.2, so we
ε
0
will have ε < 48|E|2 .
We now apply Lemma 2.1, verifying below the four necessary conditions to show that Θ
˜
˜
is (F,
L ε, d)–multiplicative. We first check that it is (E1 , ε/6, d)–multiplicative when restricted
to H G. Recall that throughout Proposition 3.1 we assume that ` ≤ 1 and `0 ≤ 1, while in
part (a) we assume furthermore that `0 L
restricts to ` on each copy of K. Let g, g 0 ∈ E1 with
0
0
g = (gx )x∈H , g = (gx )x∈H . Since E1 ⊂ E G, we may apply θb to g, g 0 , and gg 0 . Then
X

˜ B (g)θB (g 0 ), θB (gg 0 )) = 1
d(θ
d0 θb (g)θb (g 0 ), θb (gg 0 )
|B|
b∈B

1 X 0
≤κ+
d θb (g)θb (g 0 ), θb (gg 0 ) .
|B|
b∈BE

By the definitions of θb and of E1 , we realise that each component of θb (gg 0 )−1 θb (g)θb (g 0 )
is either 1 or θ(gx gx0 )−1 θ(gx )θ(gx0 ), for x ∈ E. Thus
X X

˜ B (g)θB (g 0 ), θB (gg 0 )) ≤ κ + 1
d θ(gx )θ(gx0 ), θ(gx gx0 )
d(θ
|B|
b∈BE x∈E

≤ κ + |E| ε0 ,
where in the last line we use that θ is (EG , ε0 , d)–multiplicative. Since κ + |E| ε0 < 6ε , we see
˜
that Θ is (E1 , ε/6, d)–multiplicative.
˜
The fact that the restriction to H is (E2 , ε/6, d)–multiplicative
is more straightforward.
Indeed, for h, h0 ∈ E2 we have
0
˜
d(σ(hh
), σ(h)σ(h0 )) = dHamm (σ(hh0 ), σ(h)σ(h0 )) < ε0 ,

12

BEN HAYES AND ANDREW W. SALE

where we note that we can use the multiplicative property of σ since E2 ⊆ EH .
By construction, the third condition of Lemma 2.1, bounding the distance between Θ(g, h)
and Θ(g, 1)Θ(1, h), is automatically satisfied by Θ, since these elements are equal. 
We finish part (a) by verifying the bound on d˜ Θ(1, h)Θ(g, 1), Θ(αh (g), 1)Θ(1, h) for g ∈
E1 , h ∈ E2 . We have


d˜ Θ(1, h)Θ(g, 1), Θ(αh (g), 1)Θ(1, h) = d˜ (ασ(h) (θB (g)), σ(h)), (θB (αh (g)), σ(h))

1 X 0
=
d θσ(h)−1 b (g), θb (αh (g)
|B|
b∈B

1 X 0
=
d θb (g), θσ(h)b (αh (g)) .
|B|
b∈B

0

Using Lemma 3.2, and that ` ≤ 1, we can disregard what happens for b outside of both BE
and σ(h)−1 BE for a controlled cost. This gives us the following upper bound for the above
distance:
X

1
d0 θb (g), θσ(h)b (αh (g)) .
2κ +
|B|
−1
b∈BE ∩σ(h)

BE

Since Supp(αh (g)) = h Supp(g), and E contains both Supp(g) and h Supp(g), it follows that
for every b ∈ BE ∩ σ(h)−1 (BE ) we have
Y
Y
(x)
(hx)
θσ(h)b (αh (g)) =
θσ(h)b (gh−1 x ) =
θσ(h)b (gx ).
x∈h Supp(g)

x∈Supp(g)

Note that we have used that θ(1) = 1 to restrict the number of terms in the product. We use
(hx)
(x)
that for h ∈ E (and hence for h ∈ E2 ) and b ∈ BE ∩σ(h)−1 BE we have that θσ(h)b (g) = θb (g).
Inserting this into the above equation we see that
Y
(x)
θσ(h)b (αh (g)) =
θb (gx ) = θb (g).
x∈Supp(g)

Returning to the above inequality, we have shown that
X
1
d0 (θb (g), θσ(h)b (αh (g))) = 0
|B|
−1
b∈BE ∩σ(h)

BE

so


ε
d˜ Θ(1, h)Θ(g, 1), Θ(αh (g), 1)Θ(1, h) < 2κ < .
6
This completes the proof of part (a) of Proposition 3.1.
3.4. Part (b) of Proposition
3.1. We now show that Θ is (F, c0 , d˜max )–injective, when the
L
0
length function ` on B K is `max .
In order to get (b) we will need to further restrict the size of κ (and hence also of ε0 ). We
ε
take κ small enough so that, in addition to having κ < 12
, we also have
κ<


1
min c(g), 1 | g ∈ EG \ {1} .
4

First suppose (g, h) ∈ F. If h 6= 1, then

`˜max (θB (g), σ(h)) ≥ `Hamm (σ(h)) ≥ 1 − ε0 ≥ 1/2 = c0 (g, h).

METRIC APPROXIMATIONS OF WREATH PRODUCTS

13

We may therefore assume that h = 1. Let g = (gx )x∈E . We then have that

1 X
`˜max (θB (g), 1) =
`max (θb (g))
|B|
b∈B
1 X
`max (θb (g))
≥ −κ +
|B|
b∈BE

using Lemma 3.2 to obtain the inequality. Since for b ∈ BE the components of θb (g) are either
1 or θ(gx ) for some x ∈ E, we get `max (θb (g)) = maxx∈E `(θ(gx )). Hence

|BE |
`˜max (θB (g), 1) ≥ −κ +
max `(θ(gx ))
|B| x∈E
≥ −κ + (1 − κ) max c(gx )
x∈Supp(g)

where the last inequality follows from Lemma 3.2 and the fact that θ is (EG , c, d)–injective.
By the choices of κ and EG , we get


−1
1
−κ + (1 − κ) max c(gx ) ≥
max c(gx ) + 1 −
max c(gx ) = c0 (g, 1).
4 x∈Supp(g)
4 x∈Supp(g)
x∈Supp(g)
This verifies that Θ is (F, c0 , d˜max )–injective, and thus completes the proof of Proposition 3.1.
Remark 3.3. Our proof can in fact be subtly modified to give a stronger version of Proposition
3.1, that is reminiscent of the notion of strong discrete C–approximations of Holt–Rees [HR17].
Namely, for any η > 0 we can improve the conclusion of part (b) to say that Θ is (F, c0 , d˜max )–
injective, where c0 is given by

(1 − η),
if h 6= 1
c0 (g, h) =
 max (1 − η)c(gx ), if h = 1, g = (gx )x∈H .
x∈Supp(g)

For this improved version, the parameters E, EH , EG , ε0 will depend upon η. We have elected
to not give this improved version in order to simplify the statement of the proposition and its
proof.
4. Applications of Proposition 3.1
In this section, we use Proposition 3.1 to prove Theorem 3. Part (iv) of Theorem 3 follows
immediately from Proposition 3.1, so we focus on proving the remaining three parts. Each of
parts (i),(ii),(iii) are proved below in separate subsections. We recall that the aim is to show
that, for H a countable, discrete, sofic group, the wreath product G o H is respectively sofic,
hyperlinear, or linear sofic, whenever G is such a group.
4.1. Proof of Part (i): Sofic. We restate and prove our soficity result for wreath products.
Theorem 4.1. Let G, H be countable, discrete, sofic groups. Then G o H is sofic.
Proof. In order to show that G o H is sofic, we show that G o H is C-approximable, where C is
the class of symmetric groups with the normalized Hamming distance. To do this we compose
the map Θ from Section 3.2 with a second map Ψ, as described below.
Let F ⊆ G o H be finite and ε > 0. Let EG , E, EH and ε0 > 0 be as in Proposition 3.1 for
F, ε. Define c on G \ {1} by c(g) = 12 , and so
c0 : G o H \ {1} → (0, 1/2]
as constructed in Proposition 3.1, is either 1/2 if h 6= 1, or 1/4 otherwise.

14

BEN HAYES AND ANDREW W. SALE

Since G, H are sofic we can find corresponding sofic approximations. For H we take σ : H →
Sym(B), for a finite set B, to be an (EH , ε0 )–sofic approximation; for G we take θ : G →
Sym(A), for a finite set A, to be an (EG , ε0 )–sofic approximation. Note that, since ε0 < 1/2 (see
(1) following Proposition 3.1), the (EG , ε0 )–free condition of θ implies that it is (EG , c, dHamm )–
injective.
L
With these maps, let Θ : G o H → ( B Sym(A)) oB Sym(B)Lbe the map constructed in
Section
L3, with K = Sym(A). We now explain how we embed ( B Sym(A)) oB Sym(B) into
Sym ( B A ⊕ B). First, define
!
M
M
Φ:
Sym(A) → Sym
A
B

B

by the diagonal action
Φ((πβ )β∈B ) : (aβ )β∈B 7→ (πβ (aβ ))β∈B , for πβ ∈ Sym(A), (aβ )β∈B ∈

M

A.

B

Then, use Φ to define the embedding
!
Ψ:

M

Sym(A)

!
oB Sym(B) → Sym

B

M

A⊕B

B

by
Ψ(π, τ ) : (a, b) 7→ (Φ(πτ (b) )(a), τ (b))
L L
L
for π ∈
(
Sym(A)),
τ ∈ Sym(B), a ∈
B
B
B A, and b ∈ B. A routine computation
reveals that Ψ is a homomorphism.
L
Let `0 , `max be the conjugacy-invariant length functions on B Sym(A) given by
`0 (π) = `Hamm (Φ(π)),
`max (π) = max `Hamm (πβ )
β∈B

˜ ˜
for π = (πβ )β∈B ∈
metrics as conB Sym(A). Then take d, dmax to be the bi-invariant
L
0
structed in Proposition 2.2 from the length functions
`
,
`
on
Sym(A).
max
B
L L
Because Ψ is a homomorphism, for π1 , π2 ∈ B ( B Sym(A)) , τ1 , τ2 ∈ Sym(B), we have:
˜ 1 , τ1 ), (π2 , τ2 )).
(2)
dHamm (Ψ(π1 , τ1 ), Ψ(π2 , τ2 )) = d((π
L

It thus follows directly from Proposition 3.1 that Ψ ◦ Θ is (F, ε, dL
)–multiplicative.
HammL
We now show that Ψ ◦ Θ is (F, c0 , L
dHamm )–injective. Let π ∈ B ( B Sym(A)) and τ ∈
Sym(B). Write π = (πb )b∈B for πb ∈ B Sym(A) and, for a fixed b ∈ B, let πb = (πb,β )β∈B .
For each b ∈ B such that τ (b) = b we then have
1
|{(aβ )β∈B | πb,β aβ = aβ }|
`Hamm (Φ(πb )) = 1 −
|B|
|A|
1 Y
=1−
|{a ∈ A | πb,β a = a}|
|B|
|A| β∈B
Y

=1−
1 − `Hamm (πb,β )
β∈B

which implies


X
Y

1
1 −
`˜ (π, τ ) = `Hamm (τ ) +
1 − `Hamm (πb,β )  .
|B|


b∈B
τ (b)=b

β∈B

METRIC APPROXIMATIONS OF WREATH PRODUCTS

15

Since 0 ≤ `Hamm (πb,β ) ≤ 1 we have for each b ∈ B :
Y

1 − `Hamm (πb,β ) ≤ 1 − max `Hamm (πb,β )
β∈B

β∈B

and inserting this into the above expression for d˜ shows that
˜
`((π,
τ )) ≥ `˜max ((π, τ )).
Combining equation (2) with the preceding inequality, we get for each g ∈ F \ {1}



`Hamm Ψ(Θ(g)) = `˜ Θ(g) ≥ `˜max Θ(g) ≥ c0 (g)
since Proposition 3.1 implies Θ is (F, c0 , d˜max )–injective. This shows that Ψ◦Θ is (F, c0 , dHamm )–
injective. Hence we have shown that GoH is C–approximable, where C is the class of symmetric
groups equipped with the Hamming distance and this means that G o H is sofic.

We remark that one can use the improved version of Proposition 3.1, as per Remark 3.3,
to show that Ψ ◦ Θ as considered in the above proof is an (F, ε)–sofic approximation provided
θ : G → Sym(A) and σ : H → Sym(B) are sufficiently good sofic approximations. In this way
one can in fact directly show that G o H has arbitrarily good sofic approximations.
4.2. Proof of Part (ii): Hyperlinear. In this section, we deduce hyperlinearity of G o H,
assuming that G is hyperlinear and H is sofic. Hyperlinear groups are defined by admitting
a metric approximation to unitary groups, U(n), paired with the normalized Hilbert-Schmidt
metric.
Let tr : Mn (C) → C be the normalized trace:
n

tr(A) =

1X
Ajj
n j=1

where A = (Aij ) ∈ Mn (C).
Definition 6. The normalized Hilbert-Schmidt norm on Mn (C) is defined by
kAk2 = tr(A∗ A)1/2 , for A ∈ Mn (C).
The normalized Hilbert-Schmidt metric on U(n) is therefore given by
dHS (U, V ) = kU − V k2 , for U, V ∈ U(n).
The corresponding length function is denoted `HS .
Definition 7. We say a group is hyperlinear if it is C–approximable, where C is the class of
unitary groups, paired with the normalized Hilbert-Schmidt metrics.
We will need that our approximations θ : G → U(n) not only map θ(g) far away from Id for
g 6= 1, but that in fact θ(g) is far away from the unit circle S 1 = {λ Id : |λ| = 1} in U(n). To
put this in a framework where we can take advantage of Proposition 3.1, we use the following
set-up.
Define dHS , a bi-invariant metric on U(n)/S 1 , by
dHS (U S 1 , V S 1 ) = inf 1 dHS (λU, V ), for U, V ∈ U(n).
λ∈S

Let `HS denote the corresponding length function. We will abuse this notation and write
dHS (U, V ). Note that we can directly use the normalized trace to calculate `HS (U ) as follows:
2

`HS (U )2 = inf 1 kU − λ Idk2 = inf 1 2 − 2 re(λ tr(U )) = 2 − 2 |tr(U )| .
λ∈S

λ∈S

16

BEN HAYES AND ANDREW W. SALE

In light of this, we get the following reformulation of a result of Rădulescu in [Ra08] which
gives an equivalent definition of hyperlinearity.
√
Proposition 4.2. Let G be a group and c : G \ {1} → (0, 2) any function.
Then G is hyperlinear if and only if for every ε > 0 and any finite F ⊆ G there is a positive
integer n and a function θ : G → U(n) which is (F, ε, dHS )–multiplicative and so that q ◦ θ is
(F, c, dHS )–injective, where q : U(n) → U(n)/S 1 is the quotient map.
Theorem 4.3. Let H be a countable, discrete, sofic group and G a countable, discrete,
hyperlinear group. Then G o H is hyperlinear.
Proof. We proceed in an analogous manner as for Theorem 4.1, when we dealt with soficity.
In particular, we show that G o H is C–approximable, where C is as in Definition 7. The
necessary maps to demonstrate this will be constructed as a composition, starting with Θ
from Proposition 3.1 followed by an appropriate embedding into a unitary group.
Step 1: Setting the scene.
Let F ⊆ G o H be finite and ε > 0. Let EG , E, EH and ε0 > 0 be as Proposition 3.1 for
F, ε. Let c : G \ {1} → (0, 1/2] be given by c(g) = 12 for g ∈ G \ {1} and let c0 : G o H \ {1} →
(0, 1/2] be the map constructed in Proposition 3.1. Since H is sofic we can find an (EH , ε0 )–
sofic approximation σ : H → Sym(B) for some finite set B. Since G is hyperlinear we apply
Proposition 4.2 to find an (EG , ε, dHS )–multiplicative map θ : G → U(H) for some finitedimensional Hilbert space
L H so that q ◦ θ is (EG , c, dHS )–injecitve.
Let Θ : G o H → ( B U(H)) oB o Sym(B)
 constructed from θ, σ and E in
L be the map
1
Section 3. Similarly construct Θ̄ : G o H →
U(H)/S
oB Sym(B) from q ◦ θ, σ and E.
B
Define
M
Φ:
U(H) → U(H⊗B )
B

by
O

Φ : (Vβ )β∈B 7→

Vβ , for (Vβ )β∈B ∈

M

We now define

U(H).

B

β∈B

!
Ψ:

M

U(H)

!
oB Sym(B) → U

B

M

H

⊗B



B

by

Ψ((Ub )b∈B , τ ) : (ξb )b∈B 7→ Φ(Ub ) ξτ −1 (b) b∈B

L
L L
⊗B
for (ξb )b∈B ∈
, (Ub )b∈B ∈
B H
B
B U(H), and τ ∈ Sym(B). The collection of
maps we have is summarized in Figure 2.
!
GoH

Θ

M

U(H)

!
oB Sym(B)

Ψ

U

B

B

Θ̄

!
M

U(H)/S 1

M

oB Sym(B)

B

Figure 2. A plan of the maps involved.

H

⊗B



METRIC APPROXIMATIONS OF WREATH PRODUCTS

17


˜ d˜max be the bi-invariant metrics on (L U(H)) oB Sym(B) and L U(H)/S 1 oB
Let d,
B
B
L
Sym(B),
induced by Proposition 2.2 from the length functions `0 , `max on B U(H)
L respectively,
1
and B U(H)/S , respectively, which are given by
1
`0 (V ) = `HS (Φ(V )),
2
`HS (q(Vβ ))
√
,
`max (V̄ ) = max
β∈B
2
L
L
1
for V = (Vβ )β∈B ∈
B U(H), and V̄ =√(q(Vβ ))β∈B ∈
B U(H)/S . Note that dHS is
bounded by 2, whereas dHS is bounded by 2.
Step 2: A formula for dHS (Ψ(U, τ ), Id).
We aim to bound the dHS –distance from a point in the image of Ψ to the identity in terms of
˜
the d–distance
for its pre-image. To this end, we first observe that the matrix representation
for Ψ(U, τ ) will be a block permutation matrix, with blocks corresponding to elements of B.
The matrix will have a non-zero block in the (b, b)–position precisely when τ (b) = b. Thus we
get
1 X
tr (Φ(Ub )) , where U = (Ub )b∈B .
tr (Ψ(U, τ )) =
|B|
b∈B
τ (b)=b

This implies that
2

kΨ(U, τ ) − Idk2 = 2 − 2 re(tr(Ψ(U, τ ))) = 2 −

2 X
re(tr(Ub )).
|B|
b∈B
τ (b)=b

By the definition of the Hamming metric we can rewrite the right-hand side as
2 X
1 − re(tr(Φ(Ub ))).
2`Hamm (τ ) +
|B|
b∈B
τ (b)=b

Hence
(3)

2

kΨ(U, τ ) − Idk2 = 2`Hamm (τ ) +

2 X
2
kΦ(Ub ) − Idk2 .
|B|
b∈B
τ (b)=b

Step 3: Almost multiplicativity.
√
Since kΦ(Ub ) − Idk2 ≤ 2, we can get an upper bound of
4 X
2
˜
kΨ(U, τ ) − Idk2 ≤ 2`Hamm (τ ) +
kΦ(Ub ) − Idk2 ≤ 8d((U,
τ ), 1).
|B|
b∈B
τ (b)=b

L
In summary, since Ψ is a homomorphism, for (U1 , τ1 ), (U2 , τ2 ) ∈ ( B U(H)) oB Sym(B) we
have shown
√
˜ 1 , τ1 ), (U2 , τ2 ))1/2 .
dHS (Ψ(U1 , τ1 ), Ψ(U2 , τ2 )) ≤ 2 2d((U
˜
From Proposition 3.1 we know that Θ √
is (F, ε, d)–multiplicative.
With this, the above inequality then implies that Ψ ◦ Θ is (F, 2 2ε, dHS )–multiplicative.
Step 4: Almost injectivity.
L
Let V = (Vβ )β∈B ∈ B U(H). For each β, we have that

2
`HS (q(Vβ ))
√
|tr(Vβ )| = 1 −
.
2

18

BEN HAYES AND ANDREW W. SALE

Thus
|tr(Φ(V ))| =

Y


|tr(Vβ )| ≤ 1 − max
β∈B

β∈B

`HS (q(Vβ ))
√
2

2

= 1 − `max (V̄ )2
2

2

where V̄ = (q(Vβ ))β∈B . Since 2−2 |tr(Φ(V ))| ≤ kΦ(V ) − Idk2 , we get `max (V̄ )2 ≤ 21 kΦ(V ) − Idk2 .
Inserting this into equation (3) and arguing as in Section 4.1 we see that, if Ūb = (q(Ub,β ))β∈B ,
then
2 X
2
kΨ(U, τ ) − Idk2 ≥ 2`Hamm (τ ) +
`max (Ūb )2
|B|
b∈B
τ (b)=b

2




 1 X
`max (Ūb )
≥ 2`Hamm (τ )2 + 2 

 |B|
b∈B
τ (b)=b

2





1 X
≥
`max (Ūb )
`Hamm (τ ) + |B|

b∈B
τ (b)=b

2
= d˜max (Ū , τ ), 1
where Ū = (Ūb )b∈B . As q ◦ θ is (EG , c, dHS )–injective, it follows by Proposition 3.1 that Θ̄
is (F, c0 , d˜max )–injective. Thus, for (U, τ ) in the image of Θ, it follows that (Ū , τ ) is in the
image of Θ̄, and
2
`HS (ψ(U, τ ))2 = kΨ(U, τ ) − Idk2 ≥ (c0 (x))2 .
√
Thus Ψ ◦ Θ is (F, c0 , dHS )–injective and (F, 2 ε, dHS )–multiplicative. As ε > 0 is arbitrary
the proof is complete.

As with soficity, one can use the improved version of Proposition 3.1 from Remark 3.3 to
strengthen the bounds in the above results. In particular this will show that
min `HS (Ψ(Θ(x))) ≥ 1 − ε,

x∈F \{1}

provided σ : H → Sym(B) is a sufficiently good sofic approximation and θ satisfies
min `HS (θ(g)) > 1 − κ,
g∈E

for a sufficiently large E and a sufficiently small κ. In this manner, we can directly verify the
conclusion of Proposition 4.2 for G o H if H is sofic and G is hyperlinear.
4.3. Proof of Part (iii): Linear Sofic. We recall the following definition due to Arzhantseva
and Paunescu [APa17].
Definition 8. Let F be a field. Define a bi-invariant metric drk , with corresponding length
function `rk , on GLn (F) by
1
drk (A, B) = Rank(A − B).
n
We say that a group is linear sofic over F if it is C-approximable, where C consists of all
general linear groups GLn (F), each paired with the metric drk .

METRIC APPROXIMATIONS OF WREATH PRODUCTS

19

In this section we use Proposition 3.1 to show that G o H is linear sofic if G is linear sofic
and H is sofic. Proving that the map we constructed is sufficiently injective turns out to be
trickier than in any of the other cases. As in the case of hyperlinear groups, we will need that
our linear sofic approximation θ : G → GLn (F) does not just satisfy that n1 Rank(θ(g) − Id)
is bounded away from 0 for g 6= 1, but in fact we need
1
min RankK (θ(g) − λ Id) > 0,
λ∈K× n
where RankK indicates that we are computing dimension over the algebraic closure K of F.
Thus we use the following definition.
Definition 9. Let F be a field, for A, B ∈ GLn (F), we let
drk (A, B) = min×
λ∈K

1
Rank(A − λB)
n

and `rk denote the corresponding length function.
Note that, since n1 RankF (A−B) = n1 RankK (A−B) for A, B ∈ GLn (F), we have drk (A, B) ≥
drk (A, B).
We will then use the following fact, which is a consequence of an equivalent characterization
of linear soficity given by Arzhantseva–Paunescu [APa17, Theorem 5.10].
Proposition 4.4. Let G be a linear sofic group over the field F and let K denote the algebriac
closure of F.
Then, for any δ ∈ (0, 18 ) and any finite F ⊆ G, there is a positive integer n and a function
θ : G → GLn (F) which is (F, δ, drk )–multiplicative, and so that q ◦ θ is (F, c, drk )–injective,
where c(g) = 18 − δ for all g ∈ G, and q : GLn (F) → PGLn (K) is the canoncial map given
by composing the natural inclusion GLn (F) → GLn (K) with the quotient map GLn (K) →
PGLn (K).
Proof. By [APa17, Theorem 5.10], it follows that there exists a function
θ0 : G → GLm (F)
for some m ∈ N, which is (F, δ, drk )–multiplicative and so that drk (θ0 (g) − Id) ≥
all g ∈ F \ {1}. Now consider
θ : G → GL2m (F)

1
4

− 2δ for

given in matrix block form by


θ0 (g) 0
θ(g) =
.
0
Id
Fix λ ∈ K× and g ∈ F \ {1}. If λ 6= 1, then we see that drk (θ(g), λ Id) ≥ 21 . On the other
hand, if λ = 1 then


1
1
1
1
RankK (θ(g) − λ Id) = ·
RankF (θ0 (g) − Id) ≥ − δ.
2m
2 m
8
Thus θ is the required function.



In order to use Proposition 3.1 to prove that G o H is linear sofic, we will need to use tensor
products of matrices. The main fact we will need is that if A ∈ GLn (F), B ∈ GLk (F) and
`rk (A), `rk (B) are both bounded away from zero, then `rk (A ⊗ B) is also bounded away from
zero. We formulate this precisely in the Proposition 4.7 below, whose proof uses similar ideas
to [APa17, Lemma 5.4, Prop 5.8].

20

BEN HAYES AND ANDREW W. SALE

Let Jα (A) denote the number of Jordan blocks in the Jordan normal form of A associated
to the eigenvalue α. If α is not an eigenvalue then we set Jα (A) = 0. Given a number α and a
positive integer n we let J(α, n) denote the standard n × n Jordan block with eigenvalue α. In
characteristic zero, the following is a classic result explaining how Jordan blocks behave under
tensor products, known as the Clebsch-Gordan formula (and in fact one can even say what
the precise Jordan block decomposition of J(α, n) ⊗ J(β, k) is, though we will not need this).
See, for example, [MV, Theorem 2]. For positive characteristic, this result is a consequence
of [II09, Theorem 2.2.2].
Theorem 4.5. Let K, be an algebraically closed field, α, β be nonzero elements of K, and
n, k be positive integers. Then

Jαβ J(α, n) ⊗ J(β, k) = min{n, k}.
We will use this to prove the following.
Lemma 4.6. Let K be an algebraically closed field and take A ∈ GLn (K) and B ∈ GLk (K).
Then, for each λ ∈ K,


Jλ (A ⊗ B) ≤ min k max Jα (A), n max Jβ (B) .
α∈K

β∈K

Proof. Let us first prove that Jλ (A⊗B) ≤ k maxα∈K Jα (A), as the other inequality will follow
by symmetry. First, assuming that A and B have unique eigenvalues α and β respectively,
the result of the lemma becomes
Jαβ (A ⊗ B) ≤ kJα (A).

(4)

Both sides of the above inequality are additive under taking direct sums of matrices with the
given eigenvalues. So we may assume that A and B are one Jordan block, in which case (4)
follows from Theorem 4.5.
Now suppose the eigenvalues of A and B are not necessarily unique. Since K is algebraically
closed, up to conjugacy we may write A and B as direct sums
M
M
A=
Aα , B =
Bβ
α∈F

β∈K

where Aα is the direct sum of all Jordan blocks of A associated to eigenvalue α, and similarly
for Bβ . Suppose Aα is nα × nα and Bβ is kβ × kβ . Then
M
A⊗B =
Aα ⊗ Bβ ,
α,β∈K

which leads to the following, using (4):
X
X
X
Jλ (A ⊗ B) =
Jλ (Aα ⊗ Bβ ) ≤
kβ Jα (A) ≤
kβ max Jα (A) = k max Jα (A).
αβ=λ

αβ=λ

β∈K

α∈K

α∈K

This completes the proof.



To see how the normalized rank metric drk behaves under tensor products, we remark that

Rank(A − α Id) = n − Jα (A) for every α ∈ K, implying drk (A, Id) = inf λ∈K 1 − n1 Jλ (A) .
The following is thus an immediate consequence of this fact, and of Lemma 4.6.
Proposition 4.7. Let F be a field. Let n, k ∈ N and A ∈ GLn (F), B ∈ GLk (F). Then

drk (A ⊗ B, Id) ≥ max drk (A, Id), drk (B, Id) .

METRIC APPROXIMATIONS OF WREATH PRODUCTS

21

Theorem 4.8. Let G be a linear sofic group over the field F and H be a sofic group. Then
G o H is linear sofic over F.
Proof. The structure
of the proof is analogous to that of Theorem 4.3. We compose the map
L
Θ : G o H → ( B GLn (F)) oB Sym(B) from Proposition 3.1 with a map Ψ giving us a map
from GoH to a linear group. We verify that Ψ◦Θ satisfies the required almost multiplicativity
and almost injectivity conditions.
Step 1: Setting the scene.
Recall that q : GLn (F) → PGLn (K) denotes the composition of the canonical inclusion
GLn (F) → GLn (K), where K is the algebraic closure of F, with the quotient map GLn (K) →
PGLn (K).
1
Take a finite subset F of G o H and ε > 0. Define c : G \ {1} → (0, ∞) to take the value 16
0
0
for all g 6= 1. Let EG ⊆ G, E, EH ⊆ H, c : G \ {1} → (0, ∞), and ε > 0 all be as determined
by F, ε, and c in Proposition 3.1. Note that from (1) in the proof of Proposition 3.1, we know
1
. Thus, taking δ = ε0 in Proposition 4.4 gives us a map θ : G → GLn (F)
that ε0 < 1612 < 16
0
that is (EG , ε , drk )–multiplicative and is such that q ◦ θ is (EG , c, drk )–injective.
Let σ : H → Sym(B), for some finite set B, be an (EH , ε0 )–sofic approximation and take
!
M
Θ: G o H →
GLn (F) oB Sym(B)
B

to be the map constructed from θ, σ, and E in Section 3. Meanwhile, let
!
M
Θ̄ : G o H →
PGLn (K) oB Sym(B)
B

be the map constructed using q ◦ θ in place of θ.
We now describe how to embed the image of Θ into a linear group. First define
M

Φ:
GLn (F) → GL (Fn )⊗B
B

by
Φ : (Xβ )β∈B 7→

O

Xβ , for (Xβ )β∈B ∈

M

GLn (F).

B

β∈B

Using Φ, we define
!
Ψ:

M

GLn (F)

oB Sym(B) → GL

B

M

n ⊗B

(F )



!

B

by

Ψ((Ab )b∈B , τ ) : (ξb )b∈B 7→ Φ(Ab ) ξτ −1 (b) b∈B

L 
L L
⊗B
, (Ab )b∈B ∈ B B GLn (F), and τ ∈ Sym(B).
for (ξb )b∈B ∈ B (Fn )
The collection of maps we have is summarized in Figure 3.
e d˜max be the bi-invariant metrics on the wreath products (L GLn (F)) oB Sym(B),
LetL
d,
B
and ( B PGLn (K)) oB Sym(B),
respectively,L
obtained by applying Proposition 2.2 to the
L
length functions `0 , `max on B GLn (F), and B PGLn (K), respectively, given by
`0 (X) = `rk (Φ(X)),

where X = (Xβ )β∈B

`max (X̄) = max `rk (Φ((Xβ )β∈B )),
β∈B
L
L
∈ B GLn (F), and X̄ = (q(Xβ ))β∈B ∈ B PGLn (K).

22

BEN HAYES AND ANDREW W. SALE

!
GoH

M

Θ

Ψ

oB Sym(B)

GLn (F)

GL

M

n ⊗B

(F )



!

B

B
Θ̄

!
M

PGLn (K)

oB Sym(B)

B

Figure 3. A plan of the maps involved.

Step 2: A formula for `rk (Ψ(A, τ )).
We wish to show that Ψ ◦ Θ is almost multiplicative and almost injective. To do this we
need a good handle on `rk (Ψ(A, τL
)) when (A, τ ) is in the image of Θ.
Write A = (Ab )b∈B with Ab ∈ B GLn (F). The kernel of Ψ(A, τ ) − Id is given by
 








M

 M
⊗B
ker(Φ(Ab ) − Id)
(ξb )b∈B ∈
(Fn )
: Φ(Aτ (b) )(ξb ) = ξτ (b) ⊕ 
.





b∈B
b∈B


τ (b)=b

τ (b)6=b

Focusing on the left term in the above direct sum, if we pick a cycle (b1 b2 · · · bk ) of τ , with
k ≥ 2, then ξb1 determines ξbi for i = 2, . . . , k. Thus each cycle of length greater than 1
contributes exactly n|B| to the dimension of the kernel. Let cyc0 (τ ) be the number of cycles
of length at least two in the cycle decomposition of τ . From the above discussion we see that
the dimension of ker(Ψ(A, τ ) − Id) is
X
n|B| cyc0 (τ ) +
dim(ker(Φ(Ab ) − Id)).
b∈B
τ (b)=b

It follows that
`rk (Ψ(A, τ ))

=
=

dim(ker(Ψ(A, τ ) − Id)
n|B| |B|
X 1 − `rk (Φ(Ab ))
cyc0 (τ )
1−
−
.
|B|
|B|

1−

b∈B
τ (b)=b

Since
`Hamm (τ ) = 1 −

|{b ∈ B : τ (b) = b}|
|B|

we get
(5)

`rk (Ψ(A, τ )) = `Hamm (τ ) −

cyc0 (τ )
1 X
+
`rk (Φ(Ab )).
|B|
|B|
b∈B
τ (b)=b

Step 3: Almost multiplicativity.
Equation (5) implies that
˜
`rk (Ψ(A, τ ), ) ≤ `((A,
τ )).

METRIC APPROXIMATIONS OF WREATH PRODUCTS

23

L
Bi-invariance implies that for (A1 , τ1 ), (A2 , τ2 ) ∈ ( B GLn (F)) oB Sym(B) we have:
˜ 1 , τ1 ), (A2 , τ2 )).
drk (Ψ(A1 , τ1 ), (A2 , τ2 )) ≤ d((A
˜
Thus (F, ε, drk )–multiplicativity of Ψ ◦ Θ follows from the (F, ε, d)–multiplicativity
of Θ.
Step 4: Almost injectivity.
While for almost multiplicativity we used the almost multiplicativity of Θ, for almost
injectivity we will use the almost injectivity of Θ̄.
Elementary calculations yield
`Hamm (τ ) =

2 cyc0 (τ )
|B| − |b ∈ B : τ (b) = b|
≥
.
|B|
|B|

Using this in (5), we get that
`rk (Ψ(A, τ )) ≥

1 X
1
`rk (Φ(Ab )).
`Hamm (τ ) +
2
|B|
b∈B
τ (b)=b

By repeated applications of Proposition 4.7 we have, for each b ∈ B,
`rk (Φ(Ab )) ≥ max `rk (Ab,β ).
β∈B

This implies that
1˜
`max ((Ā, τ )).
2
where Ā = ((q(Ab,β )β∈B )b∈B . If (A, τ ) lies in the image of Θ then (Ā, τ ) lies in the image of
0
Θ̄. Then, (F, c0 , d˜max )–injectivity of θ̄, coupled with the above inequality, gives us (F, c2 , drk )–
injectivity of Ψ ◦ Θ.

`rk (Ψ(A, τ )) ≥

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Ben Hayes
University of Virginia
Charlottesville, VA 22904, USA
e-mail: brh5c@virginia.edu

Andrew Sale
Cornell University
Ithaca, NY 14853, USA
e-mail: andrew.sale@some.oxon.org

