AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

arXiv:1603.00691v3 [] 29 May 2017

ALESSANDRO CARDERI AND ANDREAS THOM
Abstract. We consider the Polish group obtained as the rank-completion of an inductive limit
of finite special linear groups. This Polish group is topologically simple modulo its center, it
is extremely amenable and has no non-trivial strongly continuous unitary representation on a
Hilbert space.

Contents
Introduction
1. No non-trivial unitary representations
2. A second route to extreme amenability
3. Topological simplicity of the central quotient
4. On 1-discrete subgroups of A(q)
Acknowledgements
References

1
3
7
10
11
12
13

Introduction
Let Fq be a finite field with q = ph elements and let SLn (q) be the special linear group over
Fq . We denote by r(k) the rank of a matrix k ∈ Mn (Fq ). We equip the groups SLn (q) with the
(normalized) rank-distance, d(1, h) := n1 r(1 − h) ∈ [0, 1]. Note that d is a bi-invariant metric,
which means that it is a metric such that d(1h, 1k) = d(h, k) = d(h1, k1) for every 1, h, k ∈ SLn (q).
For every n ∈ N, we consider the diagonal embedding


1 0
 .
ϕn : SL2n (q) → SL2n+1 (q), defined by ϕn (1) := 
0 1

Observe that for every n, ϕn is an isometric homomorphism. We denote by A0 (q) the
countable group arising as the inductive limit of the family {(SL2n (q), ϕn )}n and observe that
we can extend the rank-metric d canonically to A0 (q). Let A(q) be the metric-completion of
A0 (q) with respect to d, i.e., A(q) is a Polish group and the natural extension of the rank-metric
is complete and bi-invariant. The purpose of this note is to study the topological group A(q).
Date: April 10, 2018.
1

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ALESSANDRO CARDERI AND ANDREAS THOM

In many ways one can think of A(q) as a finite characteristic analogue of the unitary
group of the hyperfinite II1 -factor, which arises as a certain metric inductive limit of the
sequence of unitary groups U(2) ⊂ U(4) ⊂ U(8) ⊂ · · · , analogous to the construction above.
However, the group A(q) reflects at the same time in an intrinsic way asymptotic properties
of the sequence of finite groups SLn (q) – in particular of the interplay of group structure,
normalized counting measure, and normalized rank-metric. In this note we want to develop
this analogy and prove various results that show similarities but also differences to the II1 factor case. Some of the techniques that we are using are inspired by the theory of von
Neumann algebras whereas others are completely independent.
Our main result is the following theorem.
Theorem. The Polish group A(q) has the following properties:
•
•
•
•

every strongly continuous unitary representation of A(q) on a Hilbert space is trivial,
the group A(q) is extremely amenable,
the center of A(q) is isomorphic to F×q and the quotient by its center is topologically simple,
it contains every countable amenable group and, in case q is odd, the free group on two
generators as discrete subgroups.

Before we start recalling some of the concepts that are used in the statement of the theorem,
let us make some further remarks. In a similar way, one can form an inductive limit of unital
rings
Fq ⊂ M2 (Fq ) ⊂ M4 (Fq ) ⊂ · · · ⊂ M0 (Fq ),
where again, the normalized rank-metric allows to complete M0 (Fq ) to a complete von
Neumann regular ring M(Fq ). Following von Neumann, we can view M(Fq ) as the coordinatization of a continuous geometry, see [27] for more details. The unit group of this ring is
naturally isomorphic to A(q). Indeed, any invertible element in M(Fq ) must be the limit of
invertible elements and a rank-one perturbation takes care of the determinant. Note that the
algebra M(Fq ) does not depend on the special choice of inclusions. Indeed, Halperin showed
in [19] that (in complete analogy to the II1 -factor situation) any choice whatsoever yields the
same algebra. We thank Gábor Elek for pointing out Halperin’s work.
Unitary representability. We recall that a group is unitarily representable if it embeds in the
unitary group of a Hilbert space as a topological group and it is exotic if every continuous
unitary representation on a Hilbert space is trivial, see [3]. While all second countable locally
compact groups are unitarily representable – via the regular representation – there are several
examples of Polish groups that are not unitarily representable. For example, the Banach space
ℓp is unitarily representable if and only if 1 ≤ p ≤ 2, [25], see also [12, 13]. The first example
of an exotic group was found by Herer and Christensen in [20] and a surprising result of
Megrelishvili, [26], states that the group of orientation preserving homeomorphisms of the
interval has no non-trivial unitary representation (not even a representation on a reflexive

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

3

Banach space). However, most of the known examples of exotic groups are either abelian or
do not have a compatible bi-invariant metric.
Amenability and extreme amenability. A topological group is said to be amenable if there exists an invariant mean on the commutative C∗ -algebra of left-uniformly continuous complexvalued functions on the group. It is a standard fact that any topological group that admits a
dense locally finite subgroup is amenable – in particular, the group A(q) is amenable for any
prime-power q. See Appendix G of [4] for more details.
A topological group is said to be extremely amenable if each continuous action of the group
on a compact topological space admits a fixed point. See [29] for more details. Let us remark
that every amenable exotic group is extremely amenable. Indeed, by amenability, any action
on a compact space preserves a probability measure; since the Koopman representation of
the action is by hypothesis trivial, the measure has to be a Dirac measure. The first example
of an extremely amenable group was found by Herer and Christensen and they proved
extreme amenability exactly by showing that the group is amenable and exotic. Nowadays,
several examples of extremely amenable groups are known, for example the unitary group of
a separable Hilbert space [18], the automorphism group of a standard probability space [14],
and the full group of a hyperfinite equivalence relation [15]. However as for the previous
property, most of the known examples do not have a compatible bi-invariant metric.
Generalizations and Open problems. The proof of the theorem is robust and can be generalized to other (inductive) sequences of non-abelian finite (quasi-)simple groups of increasing
rank. Let us finish the introduction by listing a number of open problems that we find
interesting and challenging.
Is A(q) contractible?
Is A(q)/F×q simple?
Does it have a unique Polish group topology?
More generally, is every homomorphism from A(q) to another Polish group automatically continuous?
• Does it have (isometric) representations on a reflexive Banach space?
• Does A(q)/F×q have the bounded normal generation property, see [6]?
•
•
•
•

The first question has a positive answer in the case of the hyperfinite II1 -factors by work
of Popa-Takesaki [31], but we were unable to generalize the methods to our setting. A more
detailed study of the more basic properties of the algebras M(Fq ) and their unit groups A(q)
will be subject of another study.
1. No non-trivial unitary representations
In this first section, we will prove that A(q) has no non-trivial continuous representation
on a Hilbert space, that is we will show that A(q) is exotic.

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ALESSANDRO CARDERI AND ANDREAS THOM

Definition 1.1. A complex continuous function ψ on a Polish group G is positive definite if
ψ(1G ) = 1 and for every a1 , . . . , an ∈ C and all 11 , . . . , 1n ∈ G, we have
n
X

ai ā j ψ(1−1
j 1i ) ≥ 0.

i, j=1

A positive definite function χ is a character if it is conjugation invariant, that is for every
1, h ∈ G, we have χ(h1h−1 ) = χ(1).
We will use some easy facts about positive definite functions which are covered in the
Appendix C of [4]. For example the Cauchy-Schwarz inequality for positive functions
implies that such functions have their maximal value at the identity so that any positive
definite function is bounded in absolute value by 1. We say that a positive definite function
is trivial if ψ(1) = 1 for every 1 ∈ G. Every positive definite function ψ also satisfies the
following standard inequality, which can be found in [4, Proposition C.4.2]:
(⋆)

|ψ(1) − ψ(h)|2 ≤ 2(1 − Re(ψ(1−1 h))).

There is an important relation between positive definite functions and unitary representations: the GNS construction; any positive definite function gives rise to a unitary representation and the diagonal matrix coefficients of any unitary representation are positive
definite functions. In particular, a group without any non-trivial continuous positive definite
function has no non-trivial strongly continuous unitary representation on a Hilbert space,
see Appendix C of [4] for more informations.
The following useful proposition will be needed in the course of the proof.
Proposition 1.2. Let n, m ∈ N. The canonical inclusion SL2n (q) ֒→ A0 (q) extends to an isometric
m
homomorphism SL2n (q2 ) ֒→ A0 (q).
Proof. For every k, the field Fq2k+1 is an algebraic extension of degree 2 of Fq2k , hence there
exists σk ∈ Fq2k+1 such that {1, σk } is a Fq2k -base of Fq2k+1 and there are αk , βk ∈ Fq2k such that
k+1

σ2k = αk σk + βk . For every k, and for every 1 ∈ SL2n (q2 ) there are are 10 , 11 ∈ M2n (Fq2k ) such
that 1 = 10 + σk 11 . We define


 k+1 
 k


1
β
1
0
1

k
 .
In,k : M2n q2
→ M2n+1 q2 , as In,k (1) := 
11 10 + αk 11 

It is a straightforward computation to show that for every 1, h ∈ M2n (Fq2k+1 ), we have that
In,k (1h) = In,k (1)In,k (h). Moreover, if 1 ∈ SL2n (q), then det(In,k (1)) = 1:




10 + σk 11 σk 10 + (αk σk + βk )11 
10
βk 11 



det 
 = det 

11
10 + αk 11
11 10 + αk 11 



10 + σk 11
0
= det 

11
10 + (αk − σk )11 

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

5

by hypothesis det(10 + σk 11 ) = 1 and αk − σk is the Galois-conjugate of σk , hence det(10 +
k
k+1
(αk − σk )11 ) = 1. This proves that In,k : SL2n (q2 ) → SL2n+1 (q2 ) is a well-defined group
homomorphism. The homomorphism In,k is also isometric, in fact v = v0 + σk v1 ∈ ker(1) if
and only if (v0 , v1 ) ∈ ker(In,k (1)). Composing the maps
m

Im+n−1,0 ◦ · · · ◦ In+1,m−2 ◦ In,m−1 : SL2n (q2 ) → SL2n+m (q)
m

we obtain an inclusion SL2n (q2 ) ֒→ A0 (q) that extends the standard inclusion SL2n (q) ֒→
A0 (q). This finishes the proof.

Remark 1.3. Using the idea in the proof of Proposition 1.2, one can even show that the
m
∞
inclusion SL2n (q) ֒→ A0 (q) extends to SL2n (q2 ) := ∪n SL2n (q2 ) – a special linear group over
∞
an infinite field. The characters of the group SL2n (q2 ) were completely classified by Kirillov
[21] for n > 1 and by the work of Peterson and the second author [30] for n = 1. Any
non-trivial irreducible character of SL2n (q∞ ) is induced by its center, that is, χ(1) = χ(h) for
every non-central 1, h ∈ SL2n (q∞ ). This can be used in a straightforward way to show that the
group A(q) has no non-trivial continuous character. Note that Alt(2n ) ⊂ SL2n (q) and that also
the work of Dudko and Medynets [7] can be invoked to study characters on A(q).
Now, Theorem 2.22 of [1] states that any amenable Polish group which admits a complete
bi-invariant metric is unitary representable if and only if its characters separate the points.
Whence, our observation from above implies readily that A(q) cannot be embedded into
any unitary group of a Hilbert space as a topological group. Note that only recently, nonamenable polish groups with a complete bi-invariant metric were found, which are unitarily
representable but fail to have sufficiently many characters, see [2].
In order to show that A(q) does not have any unitary representation on a Hilbert space,
we will show that every continuous positive definite function is trivial. For this, we need the
following lemma.
Lemma 1.4. Let q be a prime power and let ψ : SLn (q) → C be a positive definite function. If there
exists a non-central element 1 ∈ SLn (q) such that |1 − ψ(x−1 1x)| < ε for some ε ∈ (0, 1) and for all
x ∈ SLn (q), then
|1 − ψ(h)| < 9(2ε + 16/q)1/2 , ∀h ∈ SLn (q).
Proof. We set
χ(h) :=

X
1
ψ(x−1 hx)
|SLn (q)|
x∈SLn (q)

and note that χ : SLn (q) → C is a character. Hence, we can write
X
λπ χπ (h), h ∈ SLn (q),
χ(h) = λ +
π

where π runs through the non-trivial irreducible representations π of SLn (q), χπ denotes the
P
normalized character of π, and λ + π λπ = 1, λ ≥ 0 and λπ ≥ 0 for all π. By a result

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ALESSANDRO CARDERI AND ANDREAS THOM

of Gluck [16, Theorem 3.4 and Theorem 5.3], for every non-central element h ∈ SLn (q) and
every non-trivial, normalized, and irreducible character χπ we have |χπ (h)| < 8/q. By our
assumption |1 − χ(1)| < ε and thus
X
λπ χπ (1) > 1 − ε − 8/q.
λ = χ(1) −
π

We conclude that
|1 − χ(h)| ≤ |χ(h) − λ| + (1 − λ) ≤ 2(1 − λ) < 2ε + 16/q,

∀h ∈ SLn (q).

For fixed h, the preceding inequality, Markov’s inequality and the fact that ϕ is bounded in
absolute value by 1 imply that
n
o
|SLn (q)|
x ∈ SLn (q) |1 − ψ(x−1 hx)| ≥ 3(2ε + 16/q) ≤
3
and hence that at least 2/3 of all elements in the conjugacy class of h satisfy
|1 − ψ(k)| < 3(2ε + 16/q).
Since this holds for all conjugacy classes, we can set

A := k ∈ SLn (q) | |1 − ψ(k)| < 3(2ε + 16/q)

and conclude that |A| ≥ 2/3 · |SLn (q)|. Therefore for any h ∈ SLn (q), the set A ∩ hA−1 , ∅ and
thus there exist k1 , k2 ∈ A such that h = k1 k2 . So the inequality (⋆) yields
|ψ(k1 ) − ψ(h)|2 ≤ 2(1 − Re(ψ(k2 ))) ≤ 6(2ε + 16/q)
and hence
|1 − ψ(h)| ≤ |ψ(1) − ψ(k1 )| + |ψ(k1 ) − ψ(h)| ≤ 3(2ε + 16/q) + (6(2ε + 16/q))1/2 ≤ 9(2ε + 16/q)1/2 .
This proves the claim.



Let us now fix a positive definite function ψ : A(q) → C. Let ε > 0 and choose δ > 0 such
that d(1A(q) , x) < δ implies |1 − ψ(x)| < ε. For n large enough, the group SL2n (q) will contain
a non-central element 1 with d(1A(q) , 1) < δ. By conjugation invariance of the metric, we
conclude that |1 − ψ(x1x−1 )| < ε for all x ∈ A(q). Moreover, SL2n (q) ⊂ SL2n (qm ) ⊂ A(q) as
explained in Proposition 1.2. So applying the previous lemma to SL2n (qm ), we get that the
restriction of ψ to SL2n (q) satisfies
|1 − ψ(h)| < 9(2ε + 16/qm )1/2 ,

∀h ∈ SL2n (q).

Since this holds for any m ∈ N, we conclude that |1 − ψ(h)| ≤ 9(2ε)1/2 for all h ∈ SL2n (q) and n
large enough. Since ε > 0 was arbitrary, we must have that the restriction of ψ to any SL2n (q)
is trivial, and hence by density of A0 (q) in A(q) we can conclude the proof.

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

7

Remark 1.5. For the proof of our main theorem, it is not necessary to apply Gluck’s concrete
estimates. Indeed, it follows already from Kirillov’s work on characters of SLn (k) for infinite
fields k [21] that character values of non-central elements in SLn (q) have to be uniformly small
as q tends to infinity. This is enough to conclude the proof. One can see the existence of small
uniform bounds either by going through the techniques in Kirillov’s proof (see also the proof
of Theorem 2.4 in [30]) or by applying Kirillov’s character rigidity to a suitable ultra-product
of finite groups SLn (q) (for n fixed and q variable) which can be identified with SLn (k), where
k is the associated ultraproduct of finite fields, which is itself an infinite field.
2. A second route to extreme amenability
2.1. Lévy groups. As outlined in the beginning, every amenable and exotic group is extremely amenable. However, there is a different route to extreme amenability using the
phenomenon of measure-concentration. In this section, we want to show that A(q) is a Lévy
group and hence extremely amenable. See [29] for more background on this topic. A similar approach was recently carried out in [5] to give a direct proof that the unitary group
of the hyperfinite II1 -factor is extremely amenable. As a by-product, we can give explicit
bounds on the concentration function that are useful to give quantitative bounds in various
non-commutative Ramsey theoretic applications.
Definition 2.1. A metric measure space (X, d, µ) consists in a set X equipped with a distance
d and measure µ which is Borel for the topology induced by the metric, see [29]. In the
following we will always assume that µ is a probability measure. Given a subset A ⊂ X we
will denote by Nr (A) the r-neighbourhood of A, i.e., Nr (A) := {x ∈ X | ∃y ∈ A, d(x, y) < r}. The
concentration function of (X, d, µ) is defined as


1
(1)
α(X,d,µ) (r) := sup 1 − µ(Nr (A)) | A ⊂ X, µ(A) ≥
, r > 0.
2
Definition 2.2. A sequence of metric measure spaces (Xn , dn , µn ) with diameter constant
equal to 1, is a Lévy family if for every r ≥ 0,
α(Xn ,dn ,µn ) (r) → 0.
A Polish group G (with compatible metrid d) is called a Lévy group if there exists a sequence
(Gn )n of compact subgroups of G equipped with their normalized Haar measure µn , such
that (Gn , d|Gn , µn ) is a Lévy family. If this is the case, we say that the measure concentrates
along the sequence of subgroups.
The relationship with extreme amenability is given by the following theorem which can
be found in [29, Theorem 4.1.3].
Theorem 2.3. Every Lévy group is extremely amenable.

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ALESSANDRO CARDERI AND ANDREAS THOM

Our second route to extreme amenability is therefore to show that (SL2n (q))n is a Lev́y
family with respect to the normalized counting measure and the normalized rank-metric.
Theorem 2.4. The normalized counting measure on the groups SL2n (q) concentrates with respect to
the rank-metric. In particular, A(q) is a Lévy group and hence extremely amenable.
The proof of the theorem is a straightforward application of the following theorem, whose
proof can be found in [29, Theorem 4.5.3] or [22, Theorem 4.4]
Theorem 2.5. Let G be a compact metric group, metrized by a bi-invariant metric d, and let
{1} = H0 < H1 < H2 < · · · < Hn = G
be a chain of subgroups. Denote by ai the diameter of the homogenous space Hi /Hi−1 , i = 1, . . . , n, with
regard to the factor metric. Then the concentration function of the metric-measure space (G, d, µ),
where µ is the normalized Haar measure, satisfies


2


r
αG (r) ≤ 2exp − Pn 2  .
16 i=1 ai

Proof of Theorem 2.4. Let us fix a base {e1 , . . . , en } of Fnq . Using the previous theorem, it is
enough to show that the diameter of SLn (q)/SLn−1 (q) is smaller than n2 , where SLn−1 (q) < SLn (q)
is the subgroup fixing en . For this, it is enough to show that for every 1 ∈ SLn (q), there exists
h ∈ SLn−1 (q) ⊂ SLn (q), such that r(h1−1) ≤ 2. Let V be a 2-dimensional vector space containing
en and 1en and let V ′ be a complement. There exists h′ ∈ SL(V) with h′ (1en ) = en . Hence,
(h′ ⊕ 1V′ )1en = en and thus (h′ ⊕ 1V′ )1 ∈ SLn−1 (q). Moreover,
d(1, (h′ ⊕ 1V′ )1) =

1
2
1
r(1 − (h′ ⊕ 1V′ )1) = r((h′ − 1V ) ⊕ 0V′ ) ≤ .
n
n
n

This proves the claim.



2.2. Ramsey theory. The crucial concept behind the strategy in the previous proof is the
P
notion of length of a metric measure space, which is bounded by the value ( i a2i )1/2 as above
and essentially the infimum over these quantities. The formal definition is quite technical
and we invite the interested reader to check Definition 4.3.16 in [29]. The following lemma
is immediate from our computation.
Lemma 2.6. The length of the metric measure space (SLn (q), d, µn ) is at most 2n−1/2 .
Any estimate on the length of a metric measure space can be used directly to get explicit
bounds on the concentration phenomena, such as in the following standard lemma, whose
proof is inspired by the proof of Theorem 4.3.18 in [29].
Lemma 2.7. Let (G, d, µ) be a metric measure space of length L and let ε be a positive real. Then for
2 2
any measurable subset A ⊂ G satisfying µ(A) > 2e−ε /L we have
µ(N4ε (A)) ≥ 1 − 2e−ε

2 /L2

.

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

9

Proof. Consider the function dA : G → R defined by dA (x) := d(A, x) = inf{d(a, x) | a ∈ A} and
R
set λ := µ dA . Since dA is 1-Lipschitz, Lemma 4.3.17 in [29] implies that
µ ({x ∈ G | |dA (x) − λ| ≥ 2ε}) ≤ 2e−ε

2

/L2

.

If λ ≥ 2ε, then for every x ∈ A we have that |dA (x) − λ| = λ ≥ 2ε and the above inequality
would give us a contradiction. Therefore we must have that λ ≤ 2ε and
1 − µ(N4ε (A)) =µ ({x ∈ G | dA (x) ≥ 4ε})
≤µ ({x ∈ G | |dA (x) − λ| ≥ 2ε}) ≤ 2e−ε

2 /L2

.



Let us finish this section by applying the previously obtained bounds to deduce an explicit
metric Ramsey theoretic result for the finite groups SLn (q).
As usual, we say that a covering U of a metric space X has Lebesgue number ε > 0, if for
every point x ∈ X there exists an element of the cover U ∈ U such that the ε-neighborhood
of x is contained in U. A covering that admits a positive Lebesgue number ε will be called
a uniform covering or ε-covering. Uniform coverings have been studied in the context of
amenability, extreme amenability, and Ramsey theory in [32, 33]. Our main result in this
section is a quantitative form of the metric Ramsey property that is satisfied by the finite
groups SLn (q) – resembling the fact the A(q) is extremely amenable.
Theorem 2.8. Let ε > 0, q be a prime power, and k, m ∈ N. If we set
N := 64ε−2 max{log(2k), log(2m)}
then for any n > N and any ε-cover U of SLn (q) of cardinality at most m the following holds: for every
subset F ⊂ SLn (q) of cardinality at most k, there exists 1 ∈ SLn (q) and U ∈ U such that 1F ⊂ U.
In order to illustrate the result, consider the case k = 3, m = 2. We may think of a covering
by two sets as a coloring of SLn (q) with two colors, where some group elements get both
colors. This covering is uniform if for every element 1 ∈ SLn (q) the ε-neighborhood of 1
can be colored by one of the two colors. Now, if n > 128 · ε−2 , any subset of three elements
{a, b, c} ⊂ SLn (q) has a translate {1a, 1b, 1c} colored in the same color. This is in contrast to
(R/Z, +) equipped with its usual metric, since it is easy to see that this group does not admit
any uniform covering.
Proof of Theorem 2.8. Let U be the covering and for every U ∈ U we set U0 := {x ∈ SLn (q) |
B(x, ε) ⊂ U}. By assumption, the collection V := {U0 | U ∈ U} still forms a covering of X.
If the cardinality of the covering is at most m, there must be one element V ∈ V in the new
covering with µn (V) ≥ m1 . Take U ∈ U such that U0 = V and observe that the ε-neighborhood
2
of V is also contained in U. If n > 64 log(2m)ε−2 then 1/m > 2e−ε n/64 and hence Lemma 2.7
implies that
−

µ(U) ≥ µ(Nε (V)) ≥ 1 − 2e

ε2
16L2

ε2 n

= 1 − 2e− 64 .

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ALESSANDRO CARDERI AND ANDREAS THOM

For a subset F ⊂ SLn (q) we have
\
\

1 ∈ SLn (q) | 1F ⊂ U =
1 ∈ SLn (q) | 1h ⊂ U =
Uh−1
h∈F

h∈F

and thus if |F| ≤ k, we get that µ({1 ∈ SLn (q) | 1F ⊂ U}) ≥ 1 − 2k exp(−ε2 n/64). Hence, some
element 1 ∈ SLn (q) as desired will exist as soon as n > 64ε−2 max{log(2k), log(2m)}.

3. Topological simplicity of the central quotient
In this last section, we will determine the center of A(q) and we will prove that A(q) is
topologically simple modulo its center.
As we remarked in the introduction A(q) is isomorphic to the group of invertible elements
of M(Fq ). Therefore for every α ∈ F×q the “diagonal matrix” whose non-zero entries are equal
to α is an element of A(q) and it is in the center. Let us denote by Z ⊂ A(q) the group of
diagonal matrices with constant values, Z  F×q . We claim that Z is actually the center of A(q)
and that A(q)/Z is topologically simple. Note that the quotient A(q)/Z can be understood as
the completion of the metric inductive limit of the corresponding projective linear groups
with respect to the projective rank-metric as studied in [35].
To prove the claim, we will need the following theorem of Liebeck and Shalev which
follows from [23, Lemma 5.4], see also [35] for a discussion of the results of Liebeck-Shalev
in the context of length functions on quasi-simple groups.
Theorem 3.1 (Liebeck-Shalev). There exists c ∈ N such that for every n ∈ N and for every
1 ∈ SLn (q) with δ := min{d(1, z) | z ∈ F×q } > 0, we have CSLn (q) (1)⌈c/δ⌉ = SLn (q).
For an element 1 of the group H, we denote by CH (1) its conjugacy class. As a corollary of
this theorem we obtain the following.
Proposition 3.2. For every 1 ∈ A(q) such that δ = min{d(1, z) | z ∈ F×q } > 0, the set CA(q) (1)⌈c/δ⌉ ⊂
A(q) is dense.
Before proving the propostion, let us remark that the proposition implies our claim. Let
N < A(q) be a closed normal subgroup which contains strictly Z and take x ∈ N \ Z. By the
previous proposition, there exists m ∈ N such that CA(q) (x)m is dense, which implies that N is
dense and therefore N = A(q).
Proof of Proposition 3.2. For every ε < δ/2, we consider an element 1′ ∈ A0 (q) with d(1, 1′ ) ≤ ε.
Let n0 be such that 1′ ∈ SL2n0 (q). By Theorem 3.1 for every n ≥ n0 we have CSL2n (q) (1′ )m =
SL2n (q) for m := ⌈c/δ⌉, whence
CA(q) (1′ )m ⊃ A0 (q).
This means that for any element k ∈ A(q), there exist h1 , . . . , hm ∈ A(q) such that
′ −1
d(h1 1′ h−1
1 · · · hm 1 hm , k) ≤ ε.

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

11

Since the rank-metric is bi-invariant we obtain that
′ −1
−1
−1
′
d(h1 1′ h−1
1 · · · hm 1 hm , h1 1h1 · · · hm 1hm ) ≤ md(1, 1 ),

which implies that d(CA(q) (1)m , k) ≤ (m + 1)ε.



Remark 3.3. Since the center of A(q) is Z = F×q , we can conclude that A(q) is isomorphic to
A(q′ ) if and only if q = q′ . One could ask also if the quotient group A(q)/Z depends on q.
The question seems harder, but one can study centralizer of elements in A(q)/Z as in [37] to
deduce that this group depends at least on the characteristic of the field.
4. On 1-discrete subgroups of A(q)
In this section we want to study discrete subgroups of A(q). In fact, we will focus on
subgroups Γ ⊂ A(q) so that d(1, 1) = 1 for all non-identity elements of Γ – we call such a
subgroup 1-discrete in the metric group (A(q), d). Since the diameter of (A(q), d) is equal to 1,
1-discrete subgroups should play a special role in the study of the metric group (A(q), d). Let
us start with amenable groups.
Proposition 4.1. Every countable amenable group is isomorphic to a 1-discrete subgroup of A(q).
Proof. Let Γ be a countable amenable group and suppose at first that there is a sequence of
Følner sets (Fn ) such that Fn+1 = Fn Dn where Dn ⊂ Γ is a finite subset, so that the family
{Fn c | c ∈ Dn } consists of pairwise disjoint subsets. For every h ∈ Γ and n ∈ N, we define
Lhn := {1 ∈ Fn | h1 ⊂ Fn } and we observe that for every ε > 0 and h ∈ Γ, there exists N ∈ N such
that for every n ≥ N, we have |Lhn | > (1 − ε)|Fn |. We define elements hn ∈ M|Fn | (Fq ) as the usual
action by permutation on the left on Lhn and 0 on the complement. It is now easy to observe
that (hn )n is a Cauchy sequence for every h ∈ Γ and hence we can define a maximal discrete
embedding of Γ into the group of invertible elements of the completion of the inductive limit
of matrices of size |Fn |, which as explained in the introduction, by the result of Halperin [19],
is isomorphic to A(q).
For a general countable amenable group we will use Ornstein-Weiss theory [28]. In fact
the proposition follows from a similar argument using as Følner sets which are quasi-tiling
[28, page 24, Theorem 6]. Or one can also observe that A(q) contains the inductive limit of
symmetric groups which is the full group of the hyperfinite equivalence relation, which by
[28], contains any amenable group as maximal discrete subgroup.

The aim of this section is to show the following theorem.
Theorem 4.2. If q is odd, A(q) contains a non-abelian free group as a 1-discrete subgroup.
Theorem 4.2 follows form Elek’s work [11] and [10]. In fact, Elek showed that any amenable
skew-field embeds as a discrete sub-algebra of M(Fq ) and by [17], there exist amenable skewfields with free subgroups. However we do not need the full strength of Elek’s results and
we will construct an explicit embedding of a skew-field containing a free group into M(Fq ).

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ALESSANDRO CARDERI AND ANDREAS THOM

Note that the analogous result is not true in the setting of II1 -factors, i.e., the free group
on two generators is not a 1-discrete (which in this case should mean that different group
elements are orthogonal with respect to the trace) subgroup of the unitary group of the
hyperfinite II1 -factor. Moreover, the preceding result seems to yield the first example of a
discrete non-amenable subgroup of an amenable Polish group, whose topology is given by
a bi-invariant metric. It is an open problem, if the free group on two generators (or in fact
any discrete non-amenable group) can be a discrete subgroup of the unitary group of the
hyperfinite II1 -factor or the topological full group of the hyperfinite equivalence relation.
The proof of Theorem 4.2 is based on the following lemma, which is inspired by Elek’s
canonical rank function, see [9].
Lemma 4.3. Let Γ be a finitely generated amenable group and suppose that there is a sequence of
Følner sets (Fn ) such that Fn+1 = Fn Dn where Dn ⊂ Γ is a finite subset. Consider the embedding
Φ : Γ → M(Fq ) given in Proposition 4.1. Then for every element a of the group algebra Fq Γ which is
not a zero-divisor, the element Φ(a) ∈ M(Fq ) is invertible.
Proof. Let us fix a non zero-divisor a ∈ Fq Γ. Let us denote by S ⊂ Γ the support of a. For every
n, as in Proposition 4.1, we define Lan := {1 ∈ Fn | S1 ⊂ Fn } and we observe that for every ε,
there exists N ∈ N such that for every n ≥ N, we have |Lan | > (1 − ε)|Fn |. Observe also that a
is the limit of the elements an which acts on Lan as left translation. We claim that the vectors
{an 1}1∈Lan are linearly independent. In fact if 11 , . . . , 1k ∈ Lan and α1 , . . . , αk ∈ Fq are such that
P
P
i a(αi 1i ) = 0 then a( i αi 1i ) = 0 and whence a is a zero-divisor in Fq Γ. So the rank of an is at
least 1 − ε and hence the rank of a is 1 and therefore it is invertible.

Proof of Theorem 4.2. Let Γ be an elementary amenable, torsion free group which satisfies
the hypothesis of Lemma 4.3. By [24, Theorem 2.3], every non-zero element of the group
algebra Fq Γ is a not a zero-divisor. By Tamari [36, Example 8.16], we know that group rings
of amenable groups which do not contain zero-divisors satisfy the Ore condition, for more
about the Ore localization see [34]. Hence the Ore completion of Fq Γ, denoted by Q(Fq Γ)
is a skew-field. Consider the embedding Φ : Γ → M(Fq ) defined in Proposition 4.1 and
extend it to Fq Γ. Observe that by the universality of the Ore completion and by Lemma 4.3,
Φ extends to a map Φ : Q(Fq Γ) → M(Fq ). When Γ is also nilpotent, for example when Γ is
the Heisenberg group, Q(Fq Γ) has free subgroups by Theorem 5 of [17]. This concludes the
proof.

It is plausible that for every maximal discrete embedding of a non amenable group into
M(Fq ) the conclusion of Lemma 4.3 cannot hold.
Acknowledgements
This research was supported by ERC Starting Grant No. 277728 and ERC Consolidator
Grant No. 681207. Part of this work was written during the trimester on Random Walks and

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

13

Asymptotic Geometry of Groups in 2014 at Institute Henri Poincaré in Paris – we are grateful
to this institution for its hospitality. We thank Gábor Elek for interesting remarks that led to
the results in Section 4. We also thank the unknown referee for remarks that improved the
exposition.

References
[1] Hiroshi Ando and Yasumichi Matsuzawa. On Polish groups of finite type. Publ. Res.
Inst. Math. Sci., 48:389–408, 2012. 5
[2] Hiroshi Ando, Yasumichi Matsuzawa, Andreas Thom, and Asger Törnquist.
Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type.
arXiv:1605.06909, submitted for publication. 5
[3] Wojciech Banaszczyk. Additive subgroups of topological vector spaces, volume 1466 of
Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991. 2
[4] Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s Property (T). Cambridge
University Press, 2008. 3, 4
[5] Philip Dowerk and Andreas Thom. A new proof of extreme amenability of the unitary
group of the hyperfinite II1 factor. Bull. Belg. Math. Soc. Simon Stevin, 22(5):837–841, 2015.
7
[6] Philip Dowerk and Andreas Thom. Bounded Normal Generation and Invariant Automatic Continuity. arXiv:1506.08549, submitted for publication. 3
[7] Artem Dudko and Konstantin Medynets. On characters of inductive limits of symmetric
groups. J. Funct. Anal., 264(7):1565–1598, 2013. 5
[8] Gertrude Ehrlich. Characterization of a Continuous Geometry Within the Unit Group.
Trans. AMS, 83(2):397–416, 1956.
[9] Gábor Elek. The rank of finitely generated modules over group algebras. Proc. Amer.
Math. Soc. 131(11):3477–3485, 2011. 12
[10] Gábor Elek. Infinite dimensional representations of finite dimensional algebras and
amenability. arXiv:1512.03959. 11
[11] Gábor Elek. Convergence and limits of linear representations of finite groups. J. Algebra
450: 588–615, 2016. 11
[12] Jorge Galindo. On unitary representability of topological groups. Math. Z., 263(1):
211–220, 2009. 2
[13] Su Gao. Unitary group actions and hilbertian polish metric spaces. In Logic and its
applications, volume 380 of Contemp. Math., page 53–72. Amer. Math. Soc., Providence,
RI, 2005. 2
[14] Thierry Giordano and Vladimir Pestov. Some extremely amenable groups. C. R. Math.
Acad. Sci. Paris, 334(4):273–278, 2002. 3

14

ALESSANDRO CARDERI AND ANDREAS THOM

[15] Thierry Giordano and Vladimir Pestov. Some extremely amenable groups related to
operator algebras and ergodic theory. J. Inst. Math. Jussieu, 6(02):279–315, 2007. 3
[16] David Gluck. Sharper character value estimates for groups of Lie type. J. Algebra, 174:
229–266, 1995. 6
[17] Jairo Z. Gonçalves, Arnaldo Mandel, and Mazi Shirvani. Free products of units in
algebras. I. Quaternion algebras. J. Algebra 214:301–316, 1999. 11, 12
[18] Misha Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2
(Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., page 1–295. Cambridge
Univ. Press, Cambridge, 1993. 3
[19] Israel Halperin. von Neumann’s manuscript on inductive limits of regular rings. Canad.
J. Math., 20:477–483, 1968. 2, 11
[20] Wojchiech Herer and Jens Peter Reus Christensen. On the existence of pathological
submeasures and the construction of exotic topological groups. Math. Ann., 213(3):
203–210, 1975. 2
[21] Alexander A. Kirillov. Positive-definite functions on a group of matrices with elements
from a discrete field. Dokl. Akad. Nauk SSSR, 162:503–505, 1965. 5, 7
[22] Michel Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical
Surveys and Monographs. American Mathematical Society, Providence, RI, 2001. 8
[23] Martin W. Liebeck and Aner Shalev. Diameters of finite simple groups: sharp bounds
and applications. Ann. Math., 154(2):383–406, 2001. 10
[24] Peter A. Linnell, Noncommutative localization in group rings. Non-commutative localization in algebra and topology, Cambridge Univ. Press, Cambridge, 40–59, 2006. 12
[25] Michael G. Megrelishvili. Reflexively but not unitarily representable topological groups.
In Topology Proceedings, volume 25, 615–625 (2002), 2000. 2
[26] Michael G. Megrelishvili. Every semitopological semigroup compactification of the
group H+ [0, 1] is trivial. Semigroup Forum, 63(3):357–370, 2001. 2
[27] John von Neumann. Continuous geometry (foreword by Israel Halperin). Princeton Mathematical Series 25, 1960. 2
[28] Donald S. Ornstein and Benjamin Weiss. Entropy and isomorphism theorems for actions
of amenable groups. J. Analyse Math., 48:1–141, 1987. 11
[29] Vladimir Pestov. Dynamics of infinite-dimensional groups, volume 40 of University Lecture
Series. American Mathematical Society, Providence, RI, 2006. 3, 7, 8, 9
[30] Jesse Peterson and Andreas Thom. Character rigidity for special linear groups. J. Reine
Angew. Math. 716 (2016), 207–228. 5, 7
[31] Sorin Popa and Masamichi Takesaki. The topological structure of the unitary and
automorphism groups of a factor. Comm. Math. Phys., 155(1):93–101, 1993. 3
[32] Friedrich Martin Schneider and Andreas Thom. Topological matchings and amenability.
arXiv:1502.02293, to appear in Fundamenta Mathematicae. 9

AN EXOTIC GROUP AS LIMIT OF FINITE SPECIAL LINEAR GROUPS

15

[33] Friedrich Martin Schneider and Andreas Thom. On Følner sets in topological groups.
arXiv:1608.08185, submitted for publication. 9
[34] Bo Stenström. Rings of quotients. Die Grundlehren der Mathematischen Wissenschaften,
Band 217, An introduction to methods of ring theory, Springer-Verlag, New York, 1975.
12
[35] Abel Stolz and Andreas Thom. On the lattice of normal subgroups in ultraproducts of
compact simple groups. Proc. London Math. Soc., 108:73–102, 2014. 10
[36] Dimitri Tamari. A refined classification of semi-groups leading to generalized polynomial rings with a generalized degree concept. Proceedings of the ICM, Amsterdam 3,
439–440, 1954. 12
[37] Andreas Thom and John S. Wilson. Metric ultraproducts of finite simple groups. C. R.
Math. Acad. Sci. Paris, 352(6):463-466, 2014. 11
A.C., Institut für Geometrie, TU Dresden, 01062 Dresden, Germany
E-mail address: alessandro.carderi@tu-dresden.de
A.T., Institut für Geometrie, TU Dresden, 01062 Dresden, Germany
E-mail address: andreas.thom@tu-dresden.de

