Powers of commutators in linear algebraic groups

Abstract

Let ${\mathcal G}$ be a linear algebraic group over k , where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {\mathcal G}(k)$ . We prove that if $\gamma\in G$ such that γ is a commutator and $\delta\in G$ such that $\langle \delta\rangle= \langle \gamma\rangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.

Full text

Proceedings of the Edinburgh Mathematical Society: page 1 of 8
doi:10.1017/S0013091524000361
POWERS OF COMMUTATORS IN LINEAR ALGEBRAIC GROUPS
BENJAMIN MARTIN
Department of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
Email: B.Martin@abdn.ac.uk
(Received 26 September 2022)
Abstract: Let Gbe a linear algebraic group over k, where kis an algebraically closed field, a pseudo-
finite field or the valuation ring of a non-archimedean local field. Let G=G(k). We prove that if γ∈G
such that γis a commutator and δ∈Gsuch that hδi=hγithen δis a commutator. This generalises a
result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.
Keywords: commutator; linear algebraic group
2020 Mathematics subject classification: 20G15 (20F12; 03C98)
1. The main result
Let Gbe a group. We say that G has the Honda property1if for any γ∈Gsuch that
γis a commutator and for any δ∈Gsuch that hδi=hγi,δis also a commutator. If γ
has infinite order then the only generators of hγiare δ=γ±1, so the condition above
is only of interest when γhas finite order. The following result was proved by Honda in
1953 [8].
Theorem 1.1. Any finite group has the Honda property.
Honda’s original proof is character-theoretic. Recently, Lenstra has given a short and
elegant proof that avoids character theory completely [9].
It is natural to ask which other groups have the Honda property. Pride has given an
example of a one-relator group with torsion which does not have the Honda property
[16, Result (C), p. 488]. In this note, we extend Theorem 1.1 to certain linear algebraic
groups. By a linear algebraic group over a ring k, we mean a smooth closed subgroup
scheme of GLnfor some n∈N; if kis a field then these are just linear algebraic groups
in the usual sense [1], and they correspond to the affine algebraic groups of finite type
by [14, Corollary 4.10].
1This terminology was suggested to me by Hendrik Lenstra.
©The Author(s), 2024. Published by Cambridge University Press on Behalf of The
Edinburgh Mathematical Society. This is an Open Access article, distributed under
the terms of the Creative Commons Attribution licence (https://creativecommons.
org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduc-
tion in any medium, provided the original work is properly cited. 1
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2B. Martin
Theorem 1.2. Let Gbe a linear algebraic group over k and let G=G(k), where k is
one of the following.
(a) An algebraically closed field.
(b) A pseudo-finite field.
(c) The valuation ring of a non-archimedean local field.
Then G has the Honda property.
Remark 1.3. Let us say a group G has the strong Honda property if for any α, β ∈G
and any δ∈Gsuch that hδi=h[α, β]i, there exist σ, τ ∈ hα, βisuch that δ= [σ, τ ].
Clearly, if Ghas the strong Honda property then Ghas the Honda property and any
subgroup of Ghas the strong Honda property. Theorem 1.1 implies that any locally finite
group has the strong Honda property. In particular, if Gis a linear algebraic group over
Fpthen G(Fp) is locally finite, so G(Fp) has the strong Honda property. (Here Fpdenotes
the algebraic closure of the field with pelements.)
On the other hand, none of (P)GLn(C), (P)GLn(R), (P)SLn(C) and (P)SLn(R) has the
strong Honda property if n≥2, so Theorem 1.2(a) fails if we replace the Honda property
with the strong Honda property. To see this, consider the group Γs=ha, b |[a, b]si. Pride
(loc. cit.) shows that if 0 <t<sthen [a, b]tis not a commutator in Γsunless t≡ ±1
mod s, so Γsdoes not have the Honda property if s>6 (choose 2 ≤t≤s−2 such that t
is coprime to s). The group Γsis Fuchsian (cf. [10,§1]), so it is a subgroup of PSL2(R).
Hence PSL2(R) does not have the strong Honda property, so PSL2(C), PGL2(R) and
PGL2(C) also do not have the strong Honda property. Now choose lifts ba(resp., b
b) of a
(resp., b) to elements of SL2(R), and let b
Γs=hba,b
bi. The element [ba,b
b] has order either
sor 2s, and [ba,b
b]tis not a commutator in b
Γsif 0 < t < s and t6≡ ±1 mod s. Hence
SL2(R) does not have the strong Honda property (choose s>6 and choose 2 ≤t≤s−2
such that tis coprime to 2s), so GL2(R), SL2(C) and GL2(C) also do not have the strong
Honda property. Since we may embed SL2(R) as a subgroup of (P)GLn(C), (P)GLn(R),
(P)SLn(C) and (P)SLn(R) for any n≥3, the assertion above follows.
The proof of Theorem 1.2(a) uses the Lefschetz principle from first-order model theory.
The idea is very simple. Let Gbe a linear algebraic group over an algebraically closed
field k. For the moment, assume Gis defined over the prime field F. Set G=G(k). We
can find an F-embedding of Gas a closed subgroup of SLnfor some n: so Gis defined
as a subset of SLn(k) by polynomials over Fin n2variables and the group operations on
Gare given by polynomial maps over F. If γ, δ ∈Gthen hδi=hγiif and only if δ=γs
and γ=δtfor some s, t ∈Z. For fixed sand t, we observe that the condition
(∗)s,t for all γ, δ ∈G, if γis a commutator and δ=γsand γ=δtthen δis a commutator
is given by a first-order sentence in the language of rings involving the n2variables of
the ambient affine space. Now (∗)s,t is true if k=Fpfor any prime p, by Remark 1.3. It
follows from the Lefschetz principle that (∗)s,t is true for every algebraically closed field
k, including the characteristic 0 case. But sand twere arbitrary integers, so the result
follows. On the other hand, we see from Remark 1.3 that the analogous argument fails
for the strong Honda property, so the strong Honda property cannot be expressed in a
first-order way.
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Powers of commutators in linear algebraic groups 3
If Gis not defined over the prime field Fthen the argument above needs modification.
The trouble is that the polynomials that define Gas a closed subgroup of SLn(k) may
involve parameters from k, so (∗)s,t may fail to translate into an honest sentence. We
get around this by using a trick from [13] (cf. also the discussion after Theorem 1.1 of
[12]): first, we replace these parameters with variables, then we quantify over all possible
values of these variables. This amounts to quantifying over all linear algebraic groups of
bounded complexity in an appropriate sense. In [13,§3.2], this idea is formulated using
the language of Hopf algebras; here we give a more concrete description.
We present the details in §2. The proof of Theorem 1.2(b) follows from a closely related
argument. We prove Theorem 1.2(c) in §3.
Remark 1.4. J. Wilson gives some other first-order properties of groups that hold for
finite groups but not for arbitrary groups [19]; see [18], [5]. One can use the methods of this
paper to prove that these properties hold for linear algebraic groups over an algebraically
closed or pseudo-finite field. Likewise, Theorem 1.2 holds for definable groups in the sense
of [15, Introduction] over an algebraically closed or pseudo-finite field k; in particular, this
includes group schemes of finite type over such k. I’m grateful to Lenstra and Tiersma
for these observations.
2. Model theory and the Lefschetz principle
We give a brief review of the model theory we need to prove Theorem 1.2. For more details,
see [11]. We work in the language of rings, which consists of two binary function symbols
+ and ·and two constant symbols 0 and 1. A formula is a well-formed expression involving
+, ·, =, 0 and 1, the symbols ∨(or), ∧(and), →(implies), ¬(not), some variables and
the quantifiers ∀and ∃. For instance, (∃Y)Y2=Xand (∀X)(∃Y)(∃Z)X=Y2+Z2+ 2
and X6= 0 →X2=Xare formulas. The variable Xin the first formula is free because
it is not attached to a quantifier, while the variable Yis bound. A sentence is a formula
with no free variables; the second formula above is a sentence. For any given ring k, a
sentence is either true or false: for instance, the sentence (∀X)(∃Y)(∃Z)X=Y2+Z2+2
is true when k=Cand false when k=R. If a formula involves one or more free variables
then it doesn’t make sense to ask whether it is true or false for a particular ring k; but if
ψ(X1, . . . , Xm) is a formula in free variables X1, . . . , Xm,kis a ring and α1, . . . , αm∈k
then the expression ψ(α1, . . . , αm) we get by substituting Xi=αifor 1 ≤i≤mis either
true or false.
We can turn a formula into a sentence by quantifying over the free variables: e.g. quan-
tifying over all Xin the third formula above gives the sentence (∀X)X6= 0 →
X2=X, which is false for any field with more than two elements. Note that if
k=Rthen the expression (∀X)(∃Y)Y2=πX is not a sentence in the above sense
as it involves the real parameter π. This problem does not arise with the expression
(∀X)(∃Y)(∃Z)X=Y2+Z2+ 2: we don’t need to regard 2 as a real parameter, because
we get 2 = 1 + 1 for free by adding the constant symbol 1 to itself. More generally, if
f(X1, . . . , Xm) is a polynomial over Zin variables X1, . . . , Xmthen (say) the expression
(∃X1)· · · (∃Xm)f(X1, . . . , Xm) = 0 is a sentence.
An infinite field kis pseudo-finite if it is a model of the theory of finite fields: that is,
a sentence is true for kif it is true for every finite field. A non-principal ultraproduct of
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4B. Martin
an infinite collection of finite fields is pseudo-finite. For instance, if Uis a non-principal
ultrafilter on the set of all primes then the ultraproduct QUFpis a pseudo-finite field
of characteristic 0, and many of the infinite subfields of Fpare pseudo-finite fields of
characteristic p. See [2] and [3, (5.1)] for more details and examples.
We will use the following version of the Lefschetz principle (see [11, Corollary 2.2.9
and Corollary 2.2.10]).
Theorem 2.1 (The Lefschetz principle). Let ψbe a sentence.
(a) Let p be 0or a prime. If ψis true for some algebraically closed field of characteristic
p then ψis true for every algebraically closed field of characteristic p.
(b) ψis true for some algebraically closed field of characteristic 0if and only if
for all but finitely many primes p, ψis true for some algebraically closed field of
characteristic p.
We have an immediate corollary.
Corollary 2.2. Let ψbe a sentence. If ψis true for k=Fpfor every prime p then ψ
is true for every algebraically closed field.
Now fix a field k. Let Gbe a linear algebraic group over k. Choose an embedding of G
as a closed subgroup of SLnfor some n. We regard SLn(k) as a subset of affine space kn2
in the usual way. Denote the co-ordinates of kn2by Xij for 1 ≤i≤nand 1 ≤j≤n. Let
G=G(k). Our embedding of Gin SLnallows us to regard Gas a subset of kn2given by
the set of zeroes of some polynomials f1(Xij ), . . . , fr(Xij ) over kin the Xij.
Fix s, t ∈Z. We want to interpret the condition in (∗)s,t as a sentence. The subset
SLn(k) of kn2is the set of zeroes of finitely many polynomials over Zin the matrix
entries and the group operations on SLn(k) are given by polynomials over Zin the matrix
entries, so the conditions on α, β, γ, δ, σ, τ ∈SLn(k) that δ=γs,γt=δ,γ= [α, β ] and
δ= [σ, τ ] are given by formulas in the matrix entries of α, β, γ , δ, σ and τ. (For instance,
if we denote the co-ordinates of αby Xij then the condition α∈SLn(k) is given by the
formula det(Xij)−1 = 0.) However, the conditions α, β, γ, δ, σ, τ ∈Gmay fail to be
given by formulas as the polynomials fl(Xij ) may involve some arbitrary elements of k.
We avoid this problem as follows. Fix r≥n, let m1(Xij ), . . . , mc(Xij ) be a listing (in
some fixed but arbitrary order) of all the monomials in the Xij of total degree at most r,
and let Vrbe the subspace of the polynomial ring k[Xij ] spanned by the me(Xij ). Let
Zab for 1 ≤a≤cand 1 ≤b≤cbe variables. Define gd(Xij, Zab) = Pc
e=1 Zedme(Xij )
for 1 ≤d≤c. Now let = (ab)1≤a≤c,1≤b≤cbe a tuple of elements of k. We define
G={ηij ∈kn2|det(ηij ) = 1 and gd(ηij , ab ) = 0 for 1 ≤d≤c}.
If His a subgroup of SLn(k) defined by polynomials over kin the Xij of degree at
most rthen we say H has complexity at most r: in particular, Ghas complexity at
most r.2Conversely, any closed subgroup Hof complexity at most ris of the form Gfor
some (note that we need at most cpolynomials of the form gdto define a subgroup of
2We assume that r≥n, so the polynomial det(βij)−1 has degree n≤r.
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Powers of commutators in linear algebraic groups 5
complexity at most rbecause dim Vr=c). Any closed subgroup of SLn(k) has complexity
at most r0for some r0≥n.
Let φ(Zab) be the formula in free variables Zab given by
(∀Uij )(∀Vij )[[det(Uij ) = 1 ∧det(Vij ) = 1] ∧[gd(Uij , Zab) = 0 ∧gd(Vij , Zab)
= 0 for 1 ≤d≤c]]
→gd(Wij , Zab) = 0 for 1 ≤d≤c,
where Wij is shorthand for Pn
l=1 UilVlj . Then Gis closed under multiplication if and
only if φ(ab) is true. Likewise, there are formulas χ(Zab) and η(Zab ) such that Gis
closed under taking inverses if and only if χ(ab) is true, and the identity Ibelongs to
Gif and only if η(ab) is true.
Now consider the condition ψn,s,t,r given by
[(∀Zab)φ(Zab )∧χ(Zab)∧η(Zab )] →
(∀α∈GZ)(∀β∈GZ)(∀δ∈GZ)(δ= [α, β]s∧[α, β ] = δt)→((∃σ∈GZ)(∃τ∈GZ)δ
= [σ, τ ]),
where α= (αij ), β = (βij ), . . . are tuples representing elements of SLn(k). Here α∈GZ
is shorthand for [det(αij ) = 1] ∧hV1≤d≤cgd(αij , Zab ) = 0i, and likewise for β∈GZ, etc.
We regard ψn,s,t,r as a sentence in variables Zab,αij,βij, etc. The above discussion yields
the following: for any field kand for any n, s, t, r, the sentence ψn,s,t,r is true for kif and
only if
(†) for every closed subgroup Gof SLn(k) of complexity at most rand for every
α, β, δ ∈Gsuch that δ= [α, β]sand [α, β ] = δt, there exist σ, τ ∈Gsuch that δ= [σ, τ].
We see from the above discussion that
([)G(k) has the Honda property for every linear algebraic group Gover kif and only if
the sentence ψn,s,t,r is true for kfor all n, s, t, r.
Proposition 2.3. Let n∈N, let r≥nand let s, t ∈Z. Let k be an algebraically closed
field or a pseudo-finite field. Then ψn,s,t,r is true for k.
Proof. Fix n, s, t and r. If k=Fpfor some prime pthen G(k) has the Honda property
for every linear algebraic group Gover kby Remark 1.3, so ψn,s,t,r is true for kby ([).
Hence ψn,s,t,r is true for any algebraically closed field kby Corollary 2.2. If kis a finite
field then G(k) has the Honda property for every linear algebraic group Gover kby
Theorem 1.1, so ψn,s,t,r is true for kby ([). Hence ψn,s,t,r is true for any pseudo-finite
field. 
Proof of Theorem 1.2(a) and (b). This follows immediately from Proposition 2.3
and ([). 
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6B. Martin
Remark 2.4. We don’t know whether G(k) must satisfy the Honda property for Ga
linear algebraic group over an arbitrary field k. Tiersma has observed that if the conclusion
of Theorem 1.2 holds for a field kthen it holds for any algebraic field extension k0of k. To
see this, let G0be a linear algebraic group over k0. Let α, β ∈ G0(k0). Let γ= [α, β] and let
δ∈ G0(k0) such that hδi=hγi. There is a field k1such that k⊆k1⊆k0,k1/k is finite, G0
descends to a k1-group G1and α, β, γ , δ ∈ G1(k1). Let Gbe the Weil restriction Rk1/k(G1)
[4, A.5]. Then G(k) = G1(k1). Since G(k) satisfies the Honda property by assumption,
the result follows.
Here is another situation where we obtain a positive result. Let Gbe a connected
compact Lie group. Then every element of [G, G] is a commutator [7, Theorem 6.55],
and it follows that Ghas the Honda property. The link with linear algebraic groups
is the following: if Gis a Zariski-connected real reductive algebraic group then G:=
G(R) is compact if and only if Gis anisotropic, and in this case Gis connected in the
standard topology [1, V.24.6(c)(ii)]. A similar argument using [17, Theorem] shows that
any connected reductive linear algebraic group over an algebraically closed field has the
Honda property; this recovers a special case of Theorem 1.2(a).
3. Profinite groups
In this section, we prove Theorem 1.2(c); by a non-archimedean local field we mean
either a finite extension of a p-adic field or the field of Laurent series Fq((T)) in an
indeterminate Tfor some prime power q. In fact, we prove a more general result for
profinite groups. Recall that a topological group is profinite if and only if it is compact
and Hausdorff and has a neighbourhood base at the identity consisting of open subgroups
[6, Proposition 1.2].
Proposition 3.1. Every profinite group has the Honda property.
Proof. Let Gbe a profinite group. Let Qbe the set of finite-index normal subgroups of
G. Let γ, δ ∈Gsuch that γis a commutator and hδi=hγi. For N∈Q, let πN:G→G/N
denote the canonical projection, and set CN={(σ, τ )∈G×G|πN(δ) = [πN(σ), πN(τ)]},
a closed subset of G×G. By Theorem 1.1,CNis non-empty for each N∈Q. If
N1, . . . , Nt∈Qthen N1∩ · · · ∩ Nt∈Qand CN1∩···∩Nt=CN1∩ · · · ∩ CNt; in
particular, CN1∩ · · · ∩ CNtis non-empty. By the finite intersection property for com-
pact spaces, TN∈QCNis non-empty, so we can pick an element (σ, τ) from it. Then
πN(δ)=[πN(σ), πN(τ)] for all N∈Q. But TN∈QN= 1 as Gis profinite, so [σ, τ ] = δ
and we are done. 
Remark 3.2. Lenstra has observed that if Gis profinite then a variation of the Honda
property holds: if γ, δ ∈G,γis a commutator and the closures of hγiand hδiare equal
then δis also a commutator. The argument is very similar to the proof of Proposition 3.1.
Proof of Theorem 1.2(c). Let Fbe a non-archimedean local field with associated
norm νand valuation ring k. Fix a uniformiser π. Let Gbe a linear algebraic group over
kand let G=G(k). It is enough by Proposition 3.1 to show that Gis profinite. By
assumption, there is an embedding of Gas a closed subgroup of SLnfor some n. So G
is a subgroup of SLn(k) and there are polynomials f1, . . . , ftin n2variables over kfor
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Powers of commutators in linear algebraic groups 7
some t∈N0such that G={aij ∈kn2|fi(aij) = 0 for 1 ≤i≤t}(we need only finitely
many polynomials as discrete valuation rings are Noetherian). The operations of addition
and multiplication on kare continuous; it follows easily that Gis a closed subspace
of kn2with respect to the topology on kn2induced by ν, and the group operations
on Gare continuous. Hence Gis a compact topological group. The open subgroups
Gn:= {g∈G|g= 1 mod πn}form a neighbourhood base at the identity. It follows
that Gis profinite, as required. 
Acknowledgements. I’m grateful to Hendrik Lenstra for introducing me to the
Honda property, for stimulating discussions and for sharing a draft of his preprint [9]
with me. I’m grateful to Hendrik and to Samuel Tiersma for comments on earlier drafts
of this note. I’d also like to thank the referees for some comments and corrections, and
the Edinburgh Mathematical Society for supporting Lenstra’s visit to Aberdeen.
Competing interests. The author declares none.
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