arXiv:1802.07109v1 [math.LO] 20 Feb 2018

THE DERIVED SUBGROUP OF LINEAR AND
SIMPLY-CONNECTED O-MINIMAL GROUPS
ELÍAS BARO
Abstract. We show that the derived subgroup of a linear definable
group in an o-minimal structure is also definable, extending the semialgebraic case proved in [16]. We also show the definability of the derived
subgroup in case that the group is simply-connected.

1. Introduction
Let R = hR, <, +, ·, · · · i be an o-minimal expansion of a real closed field
R. Algebraic groups over R are clearly definable in R; on the other hand, if
G is a group definable in R and R = R then G has a Lie group structure (see
Preliminaries). In fact, the behaviour of o-minimal groups rests in between
algebraic groups and Lie groups. The definability of the derived subgroup
is a good example of this dichotomy.
As it is well-known, the derived subgroup of an irreducible algebraic group
is an irreducible algebraic subgroup. In the context of Lie groups, the derived
subgroup is a virtual subgroup (i.e., the image of a Lie homomorphism).
However, there are examples of Lie groups –even solvable– whose derived
subgroup is not closed [14, Ex.1.4.4]. In two important situations it is closed:
either if the Lie group is linear or it is simply-connected. In both cases the
proof relies on Lie’s third fundamental theorem and therefore it cannot be
reproduced in the o-minimal setting.
A. Conversano [7, §1] showed an example of an o-minimal group G whose
derived subgroup G′ is not definable (remarkably, the example is semialgebraic over R). Thus, the situation concerning the derived subgroup of
o-minimal groups could seem closer to Lie groups rather than to the algebraic ones. However, Conversano’s example is a central extension of a simple
group and therefore it is not solvable. Surprisingly, in [3] we proved that if
G is a solvable connected o-minimal group then G′ is definable. Moreover,
the commutator width of G is bounded by dim(G). Recall that the derived
(or commutator) subgroup of G is
[
G′ =
[G, G]n
n∈N

Date: February 21, 2018.
1991 Mathematics Subject Classification. 03C64.
Key words and phrases. o-minimal group, simply-connected, commutator.
Research partially supported by the program MTM2017-82105-P.
1

where [G, G]n denotes the definable set of at most n products of commutators. The commutator width is the smallest n ∈ N such that G′ = [G, G]n
in case it exists.
In this paper we prove that if G is a connected o-minimal group and G is
either linear (Theorem 3.1) or simply-connected (Theorem 4.8) then G′ is
definable.
In Section 3 we address the linear case. A. Pillay already showed in [16]
that if G is semialgebraic and linear then G′ is semialgebraic. He avoids the
use of Lie’s third fundamental theorem by considering the Zariski closure.
We will combine the result of Pillay with the definability of the derived
subgroup in the solvable case established in [3]. Furthermore, we will also
provide a bound of the commutator width.
In Section 4 we make use of the developments of o-minimal homotopy in [1]
to show that a normal connected definable subgroup of a simply-connected
definable group is simply-connected (Proposition 4.3). This allows us to
make induction arguments, so we can apply the strategy used in [3] to reduce
the problem to a minimal configuration: a central extension of a semisimple
group (Proposition 4.6).
Finally, in Section 5 we apply our results to prove an o-minimal version
of a classical result by A. Malcev (Theorem 4.6) concerning the existence of
cross-section of projection maps of quotients of simply-connected Lie groups.
2. Preliminaries
We fix an o-minimal expansion R of a real closed field R. Henceforth
definable means definable in the structure R possibly with parameters. Let
G be a definable group in R, we refer to [15] for the basics on o-minimal
groups.
For any fixed p ∈ N, the group G is a topological group with a definable
C p -manifold structure compatible with the group operation. Any definable
subgroup of G is closed and a C p -submanifold of G. Since R expands a field,
we have elimination of imaginaries and therefore the quotients of definable
groups by definable subgroups are again definable. A definable group G is
linear if G ≤ GL(n, R) for some n ∈ N.
Any definable subset X of G is the disjoint union of finitely many connected definable components, i.e. definable subsets which cannot be written as the union of two proper open definable subsets. In particular, the
connected component Go of G which contains the identity is a –normal–
subgroup of finite index (it is the smallest one with that property). We say
that G is connected if Go = G. Moreover, the group G has the descending
chain condition on definable subgroups (dcc for short): any strictly descending chain of definable subgroups –which must be closed in the topology– of
G is finite.
In [17] the authors define the Lie algebra of G similarly as in the classical
case. We define the tangent space Te (G) as the set of all equivalence classes
2

of definable C 1 -curves σ : (−1, 1) → G with σ(0) = e, where two curves are
equivalent if they are tangent at 0. We denote by σ̄ the equivalence class
of σ and we endow Te (G) with the natural vector space structure as in the
classical case. Given a local C 3 -chart ϕ : U → Rn around e ∈ U ⊆ G,
ϕ(e) = 0, we can identify Te (G) with Rn via the isomorphism sending σ
to (ϕ ◦ σ)′ (0). Moreover, since ϕ(ey) = ϕ(y) and ϕ(xe) = ϕ(x), using the
Taylor expansion we get that
ϕ(xy) = ϕ(x) + ϕ(y) + α(ϕ(x), ϕ(y)) + · · ·

where α is a bilinear vector-valued form and dots stands for elements of
order greater than 2. The transposition of x and y yields
ϕ(yx) = ϕ(y) + ϕ(x) + α(ϕ(y), ϕ(x)) + · · ·

and therefore we get that
(1)

ϕ([x, y]) = ϕ(x−1 y −1 xy) = γ(ϕ(x), ϕ(y)) + · · ·

where γ(ϕ(x), ϕ(y)) = α(ϕ(x), ϕ(y)) − α(ϕ(y), ϕ(x)) and dots stand for the
terms of order greater then 2. It turns out that Te (G) with the bracket
operation [X, Y ] := γ(X, Y ) is the Lie algebra of G, denoted by Lie(G)
–and which do not depend on the chart ϕ chosen.
Many basic results from the Lie theory have an o-minimal analogue. For
example, if H is a definable subgroup of G then Lie(H) is a Lie subalgebra
of Lie(G). Furthermore:
Fact 2.1. [17, Claim 2.20] If G1 and G2 are two connected definable subgroups of a definable group G with the same Lie algebra then G1 = G2 .
In some sense, locally definable groups play the
S role of virtual Lie groups.
A locally definable group [8] is a subset G = n∈N Xn of Rℓ which is a
countable union of increasing definable subsets Xn and whose group operation restricted to Xn × Xn is contained in Xm for some m ∈ N and it is
a definable
map. A homomorphism
f : G → H of locally definable groups
S
S
G = n∈N Xn and H = m∈N Ym is a locally definable homomorphism if
for each n there is m such that f (Xn ) ⊆ Ym and the restriction f ↾ Xn is
definable.
As before, G has a locally definable C p -manifold structure, a submanifold
of GL(n, R) in case that the group G is linear. We say that a subset Y ⊆ G
is compatible if Y ∩ Xn is definable for each n ∈ N. Any compatible subset Y
is a disjoint union of its countable many compatible connected components,
and we say G is connected if it is equal to the connected component G o of
the identity.
Fact 2.2. Let G be a connected definable group and let g be its Lie algebra.
Then G′ is a connected locally definable subgroup such that [g, g] ⊆ Lie(G′ ).

Proof. Since G × G → G : (x, y) 7→ [x, y] = x−1 y −1 xy is continuous we
get that [G, G]1 is a connected definable subset of G. Therefore [G, G]n =
n
[G, G]· · ·[G, G] is connected and definable for each n ∈ N. In particular,
3

S
G′ = n∈N [G, G]n is locally definable because [G, G]n [G, G]n = [G, G]2n
for each n ∈ N. On the other hand, [G′ ]o ∩ [G, G]n is an open and closed
definable subset of [G, G]n and therefore [G, G]n ⊆ [G′ ]o , so that [G′ ]o = G′ .
Next, let us see that [g, g] ⊆ Lie(G′ ), we follow [14, §4]. It suffices to
show that given X, Y ∈ g then [X, Y ] ∈ Lie(G′ ). Let x, y : (−1, 1) → G be
definable C 1 -curves such that x′ (0) = X and y ′ (0) = Y . Then
(
√
√
[x( t), y( t)], if t ∈ [0, 1),
p
p
c(t) :=
−1
[x( |t|), y( |t|)] , if t ∈ (−1, 0],
is a definable C 1 -curve such that c′ (0) = [X, Y ]. Indeed, by equation (1)
above,
√
√
√
√
[x( t), y( t)] = γ(x( t), y( t)) + · · ·
where “ ” denotes the image by the chart ϕ. Thus,
√
√
√
√
lim 1t [x( t), y( t)] = lim 1t γ(x( t), y( t))
t→0+

=

t→0+

√
√
lim γ( x(√tt) , y(√tt) )
t→0+

= γ(X, Y ) = [X, Y ].

On the other hand,
lim

t→0−

=

p

1
t [x(

p
|t|), y( |t|)] = lim

t→0−

√
√
−t) y( −t)
lim −γ( x(√−t
, √−t )
t→0−

√

1
t γ(x(

√
−t), y( −t))

= −γ(X, Y ) = −[X, Y ]

p
p
and therefore limt→0− 1t [x( |t|), y( |t|)]−1 = [X, Y ], as required.



A Lie subalgebra of gl(n, R) is said to be algebraic if it is the Lie algebra
of an algebraic subgroup of GL(n, R). Given a Lie subalgebra g of gl(n, R),
a(g) denotes the minimal algebraic Lie subalgebra of gl(n, R) containing g.
We recall that if g is a subalgebra of gl(n, R) then [g, g] = [a(g), a(g)] is
algebraic (see [14, Ch.3, §3]).
If G is a semialgebraic subgroup of GL(n, R) then G and its Zariski closure
G in GL(n, R) have the same dimension. This is a crucial aspect in the proof
of the following result (the bound in the commutator width can be deduced
from the proof there noting that [G, G]−1
n = [G, G]n for each n ∈ N):
Fact 2.3. [16, Cor.3.3] Let G be a semialgebraic subgroup of GL(n, R). Then
G′ is semialgebraic and its commutator width is bounded by dim(G).
It is not longer true in general that if G is a linear o-minimal group then
dim(G) = dim(G). Nevertheless, in the proof of Theorem 3.1 we will make
′
use of the fact that G and G still have an strong relation precisely via G
(as pointed out in [19]). We recall that by [19, Lem.2.4] the Zariski Lie
algebra of an algebraic subgroup of GL(n, R) and its o-minimal Lie algebra
canonically coincides. Moreover:
4

Fact 2.4. [2, Prop.3.9] Let G be a definable subgroup of GL(n, R) and G its
Zariski closure. Then, Lie(G) = a(Lie(G)). Furthermore, if G is connected
then G is irreducible and G is normal in G.
The solvable radical R(G) of a definable group G is the maximal normal
solvable connected definable subgroup of G. We say that G is semisimple if
R(G) is trivial. A non-abelian definable group G is definably simple if there
is no infinite normal definable subgroup of G. We recall a basic result of
semisimple groups, the symbol ≃ denotes definably isomorphic.
Fact 2.5. [11] Let G be a connected definable group. Then G is semisimple
if and only if its Lie algebra g is semisimple. In this case, we have that
G′ = G and there are definably simple subgroups C1 , . . . , Cℓ such that G ≃
C1 × · · · × Cℓ .
We finish with some topological remarks on o-minimal groups. Given a
connected definable group G in the o-minimal structure R, we define the
o-minimal n-homotopy group πnR (G) as in the classical case via definable
maps and definable homotopies pointed in the identity element [1, §4]. We
say that G is simply-connected if π1R (G) = 1.
If R1 is an elementary extension of R and G(R1 ) is the realization of G in
R1 , then πnR1 (G(R1 )) and πnR (G) are canonically isomorphic, so henceforth
we shall omit the superscript and we will write πn (G). For, it can be deduced
from the following stronger result which will be crucial in our work:
Fact 2.6. [1, Thm.3.1, Cor.4.4] Let X ⊆ Rn and Y ⊆ Rm be connected
semialgebraic sets defined over Q. Then every continuous map f : X → Y
definable in R is definably homotopic to a semialgebraic map g : X → Y
defined over Q. If g1 , g2 : X → Y are two continuous semialgebraic maps defined over Q which are definably homotopic, then they are semialgebraically
homotopic over Q.
In particular, the o-minimal n-homotopy group πn (X) is canonically isomorphic to the classical homotopy group πn (X(R)) of the realization of X
in the real numbers.
Similarly, we can define the o-minimal n-homotopy group of a locally
definable group G and again we have invariance under elementary extensions.
As it happens with Lie groups, the fundamental group interacts with map
coverings: an onto locally definable homomorphism p : G → H is a locally
definable
S covering if there is a family of open definable subsets {Uj }j∈J of
H = n∈N Hn whose union is H, each Hn is contained in the union of finitely
many Uj , and each p−1 (Uj ) is a disjoint union of open definable subsets of
G each of which is mapped homeomorphically onto Uj .
Fact 2.7. Let G and H be connected locally definable groups and let f : G →
H be a locally definable surjective homomorphism. If dim(ker(f )) = 0 then
f is an isomorphism.
5

Proof. Since dim(ker(f )) = 0, the map f is a locally definable covering [8,
Thm.3.6]. Therefore, by [8, Prop.3.4 and 3.12],
ker(f ) = π1 (H)/f∗ (π1 (G)),
where f∗ : π1 (G) → π1 (H) : [γ] 7→ [f ◦ γ]. Since π1 (H) = 1 we get that
ker(f ) = 1, as desired.

3. Linear groups
If G is a connected linear Lie subgroup of GL(n, R) then G′ is a closed
subgroup. Indeed, if g denotes the Lie algebra of G then we have that G′
is a connected virtual subgroup of G whose Lie algebra is [g, g]. On the
other hand, [g, g] is the Lie algebra of an algebraic subgroup H of GL(n, R).
Therefore, G′ and [H ′ ]o have the same Lie algebra, so they are equal (see
Ch.1 §2 and Ch.4 §1 in [14]). It follows that G′ is closed in G.
We cannot adapt the above argument to prove that if G is a linear ominimal group then G′ is a definable subgroup. Though G′ is a connected
locally definable subgroup of G, it is not true that locally definable subgroups
of G are uniquely determined by their Lie algebra. For example, the group
R and its finite elements Fin(R) have the same Lie algebra.
Theorem 3.1. Let G ≤ GL(n, R) be a connected definable group in R.
Then G′ is semialgebraic and connected. Moreover, the commutator width
of G is bounded by dim(G) + dim(G′ ) − dim(G′′ ).
Proof. We argue by induction on the dimension, the initial step is trivial.
So assume that dim(G) > 0.
Let g be the Lie algebra of G. By Fact 2.4 the Zariski closure H := G ≤
GL(n, R) of G is an irreducible algebraic subgroup of GL(n, R) whose Lie
algebra h := Lie(H) equals a(g). The derived subgroup H ′ of H, is also
an irreducible algebraic group with Lie(H ′ ) = [a(g), a(g)] = [g, g]. Denote
G1 := H o and G2 := [H ′ ]o , which are connected semialgebraic subgroups of
GL(n, R). Since Lie(G2 ) = [g, g] ⊆ g it follows from Fact 2.1 that
G2 E G E G1 .
We prove that G′ equals the connected semialgebraic group G2 . By Fact
2.3 the groups G′1 and G′2 are both semialgebraic and connected. Thus, the
quotient G1 /G2 is abelian since G′1 = [G1 , G1 ] = [H o , H o ] E [H, H]o = G2 .
In particular G/G2 is abelian, so that G′ ≤ G2 .
On the other hand, consider the connected definable group G/G′2 and
note that it is non-necessarily linear. However, it is solvable. Indeed, we
already showed above that G′1 E G2 , so that [G1 /G′2 ]′ = G′1 /G′2 E G2 /G′2 is
abelian. Then [G/G′2 ]′ E [G1 /G′2 ]′ is abelian and therefore G/G′2 is solvable,
as desired. Thus, by [3, Thm.3.1] we deduce that [G/G′2 ]′ = G′ /G′2 is
definable and connected, and the commutator width of G/G′2 is bounded by
dim(G/G′2 ). In particular, G′ is definable and connected.
6

Finally, by Fact 2.2 we have that
Lie(G2 ) = Lie([H ′ ]o ) = Lie(H ′ ) = [g, g] ⊆ Lie(G′ )

and therefore G′ = G2 by Fact 2.1. Note that G′′ = G′2 and thus by the above
we have that the commutator width of G/G′′ is bounded by dim(G/G′′ ) =
dim(G) − dim(G′′ ). On the other hand, by Fact 2.3 the commutator width
of G′ = G2 is bounded by dim(G′ ). Hence, the commutator width of G is
bounded by dim(G) − dim(G′′ ) + dim(G′ ), as required.

We complete the linear case by considering non-connected definable linear
groups:
Corollary 3.2. Let G be a linear group definable in an o-minimal structure, and A and B be two definable subgroups which normalize each other.
Then the subgroup [A, B] is definable and [A, B]o = [Ao , B][A, B o ]. Furthermore, any element of [A, B]o can be expressed as the product of at most
dim([A, B]o ) commutators from [Ao , B] or [A, B o ] whenever Ao or B o is
solvable.
Proof. It suffices to prove that H := Ao B o satisfies condition (∗) of [3,
Thm.3.1]. That is, if K is a normal definable subgroup of H such that H/K
is the central extension of a definable simple group then (H/K)′ = H ′ K/K is
definable. Since H is linear and connected, the latter follows from Theorem
3.1.

4. Simply-connected definable groups
A. Malcev proved the existence of cross-sections of quotients of simplyconnected Lie groups by normal closed subgroups. This is a key result that,
for example, it allows to study central extensions of simply-connected Lie
groups via analytic sections [12].
Fact 4.1. [13] Let G be a simply-connected Lie group and let H E G be a
closed connected subgroup. Let π : G → G/H be the natural homomorphism.
Then there exists an analytic mapping σ : G/H → H such that π ◦ σ = id.
Note that with the above notation,
G → (G/H) × H
x 7→ (π(x), x−1 σ(π(x)))

is a homeomorphism and therefore both H and G/H are simply-connected.
We are interested in an o-minimal version of this consequence because it will
allow us to make arguments by induction. However, the proof in [13] goes
through the 1-1 correspondence between Lie algebras and simply-connected
Lie groups, which it is not available in the o-minimal context.
We follow another approach. E. Cartan proved in [6] that any connected
Lie group has trivial second homotopy group. His proof again goes through
Lie’s third fundamental theorem. W. Browder [5] later gave an alternative
proof –using just homological methods– which is also valid for H-spaces
7

with finitely generated homology. Recall that a topological space X is an
H-space if there exists a continuous map f : X × X → X and an element
e ∈ X such that both f (−, e) and f (e, −) are homotopic to the identity map
id : X 7→ X.
Lemma 4.2. Let G be a connected definable group G. Then π2 (G) = 0.
Proof. By the Triangulation theorem we can assume that there is a finite
simplicial complex K with vertices over Q such that G = |K|(R), where
|K|(R) denotes the realization of K in R. Moreover, we can assume that
the identity of G is one of the vertices.
By Fact 2.6, the group operation on |K|(R) is definably homotopic to
a continuous semialgebraic map f : |K|(R) × |K|(R) → |K|(R) which
is defined over Q. Furthermore, both f (−, e) and f (−, e) are clearly definably homotopic to the identity map id, so again by Fact 2.6 both are
also semialgebraically homotopic to the id over Q. Thus, we can consider f R : |K|(R) × |K|(R) → |K|(R), the realization of K and f over
the real numbers. The polyhedron |K|(R) with the map f R is an H-space.
Moreover, since K is a finite simplicial complex the homology groups of
|K|(R) are clearly finitely generated. Thus, by [5, Thm.6.11] we have that
π2 (|K|(R)) = 0 and in particular π2 (G) = 0, as required.

A continuous definable map p : E → B is a definable fibration if p has the
homotopy lifting property with respect to all definable sets, i.e. for every
definable set X, for every definable homotopy H : X × I → B and for every
definable map g : X → E such that p ◦ g = H(−, 0) there is a definable
homotopy H1 : X × I → E such that p ◦ H1 = H and H1 (−, 0) = g(−).
With the above lemma and the fact that the projection map of quotients of
definable groups are definable fibrations we get:
Proposition 4.3. Let G be connected definable group, and let H be a normal
connected definable subgroup of G. Then G is simply-connected if and only
if both H and G/H are simply-connected.
Proof. By [4, Cor.2.4] the projection map G → G/H is a definable fibration.
Therefore, by [1, Thm.4.9], for each n ≥ 2, the o-minimal homotopy groups
πn (G, H) and πn (G/H) are isomorphic. In particular, we have the following
long exact sequence via the o-minimal homotopy sequence of the pair (G, H),
see [1, §4],
π2 (G/H) → π1 (H) → π1 (G) → π1 (G/H) → 0.
Since by Lemma 4.2 we have that π2 (G/H) = 0, we obtain the exact sequence
0 → π1 (H) → π1 (G) → π1 (G/H) → 0.

Therefore, π1 (G) = 0 if and only if both π1 (H) = 0 and π1 (G/H) = 0, as
required.

8

Once we have that normal connected definable subgroups and their quotients are also simply-connected, we will be able to make induction arguments. For example, we have the following consequence analogue to the
classical one:
Corollary 4.4. Let G be a connected solvable definable group. Then the
following are equivalent:
(1) G is torsion-free.
(2) G is definably diffeomorphic to Rdim(G) .
(3) G is simply-connected.
In particular, if G is simply-connected then any connected definable subgroup
is simply-connected.
Proof. 1) implies 2) follows from [20, Cor.5.7] and 2) implies 3) is obvious.
Let us prove by induction on the dimension that 3) implies 1), the initial
case is obvious. Since G is solvable, by [3, Thm.4.1] we have that G′ is
a normal connected definable proper subgroup of G. If G′ 6= 1 then by
Proposition 4.3 and by the induction hypothesis we get that both G′ and
G/G′ are torsion-free. In particular, G is also torsion-free.
Thus, we can assume that G is abelian. We consider the definable homomorphism fn : G → G : g 7→ gn for each n ∈ N. Note that ker(fn ) < G by
[21, Prop.6.1]. If 1 < ker(fn )o < G for some n ∈ N, then again by Proposition 4.3 and by induction we have that both ker(fn )o and G/ ker(fn )o are
torsion-free. In particular, G is also torsion-free. Hence, we can assume that
ker(fn )o = 1 for all n ∈ N. Thus, by Fact 2.7, we also have that ker(fn ) = 1
for all n ∈ N, as required.
Finally, suppose that G is simply-connected, and let H be a connected
definable subgroup of G. By the above equivalences, we have that G is
torsion-free, so that H is also a connected torsion-free solvable definable
group. Thus, H is simply-connected, as required.

In order to prove the definability of the commutator subgroup of simplyconnected groups we will be concerned with the following configuration: a
definable central extension of a semisimple definable group. These extensions were profoundly studied in [11]:
Fact 4.5. [11, Cor.5.3] Let G be a connected central extension of a semisimple definable group. Then for each n, the set Z(G) ∩ [G, G]n is finite.
Proposition 4.6. Let G be a simply-connected definable group such that
R(G) = Z(G)o . Then G′ is definable and simply-connected.
Proof. By Fact 2.2 the derived subgroup G′ of G is a connected locally
definable group of G and the projection map
π ↾G′ : G′ → G/Z(G)o

is clearly a locally definable homomorphism. Since R(G) = Z(G)o , the connected definable group G/Z(G)o is semisimple and therefore [G/Z(G)o ]′ =
G/Z(G)o by Fact 2.5, so that π ↾G′ is surjective.
9

Moreover, by Fact 4.5 the compatible subgroup ker(π ↾G′ ) = G′ ∩ Z(G)o
of G′ has dimension 0. Therefore, since G/Z(G)o is simply-connected by
Proposition 4.3, it follows from Fact 2.7 that π ↾G′ is a definable isomorphism, as required.

We already have all the ingredients to prove the definability of the derived
subgroup of simply-connected definable groups.
Fact 4.7. [3, Cor.4.3] Let G be a definable group and let A, B be normal
connected definable subgroups of G with [A, B] ≤ Z(B) or [A, B] ≤ Z(A).
Then [A, B] is definable and connected.
Proof. For the sake of the presentation, we include a proof in case that
[A, B] ≤ Z(B), the other one is similar. For any a ∈ A and b1 , b2 ∈ B
we have [a, b1 b2 ] = [a, b2 ][a, b1 ]b2 = [a, b2 ][a, b1 ] = [a, b1 ][a, b2 ]. Thus, the
set [a, B]1 is a group, which is also definable and connected since it is the
image of the continuous map B → B : b 7→ [a, b]. In particular, for any
a1 , . . . , aℓ ∈ A we have that
[a1 , B]1 · · · [aℓ , B]1

is a connected definable subgroup of B. Therefore [A, B] equals any such
finite product of maximal dimension, so it is definable and connected.

Theorem 4.8. Let G be a simply-connected group definable in an o-minimal
structure, A and B be two normal connected definable subgroups of G. Then
[A, B] is a normal connected definable subgroup of G.
Proof. Let G be a potential counterexample to our statement of minimal
dimension. Note that dim(G) > 2 because otherwise G is abelian. Let A
and B be two normal connected definable subgroups of G for which [A, B]
is either non-definable or definable but non-connected, and with
d := min(dim(A), dim(B)) ≥ 1

minimal. By Proposition 4.3 the normal definable subgroup AB of G is
simply-connected and therefore G = AB. Note that since A and B are
normal in G we have that [A, B] E A ∩ B.
Claim 1. There is not a normal connected definable subgroup C of G contained in A ∩ B with dim(C) < d and C  Z(A) ∩ Z(B).
Proof. Suppose there exists such a subgroup C, and say it does not centralize
B. Thus, [C, B] is a non-trivial normal connected definable subgroup of G
because dim(C) < d.
Notice that [C, B] is normal in G and therefore by Proposition 4.3 we have
that G/[C, B] is simply-connected. Denote by “ ” the quotients by [C, B].
Since G was the minimal counterexample, we get that [A, B] is definable and
connected. But clearly [A, B] = [A, B] = [A, B]/[C, B] since [C, B] ≤ [A, B],
and it follows that [A, B] is definable and connected, a contradiction.

Claim 2. We may assume A = A ∩ B  Z(A) ∩ Z(B).
10

Proof. If (A ∩ B) ≤ Z(A) ∩ Z(B) then [A, B] ≤ Z(A) ∩ Z(B). By Fact 4.7
we obtain that [A, B] is definable and connected, a contradiction. Hence,
we have that A ∩ B  Z(A) ∩ Z(B).
On the other hand, by Claim 1 we get that dim(A∩B)o = dim(A∩B) = d.
Since A and B are definably connected it follows that A ∩ B equals A or B,
say A.

In particular, we are now in the situation in which A E B = G.
Claim 3. The subgroups A and B are equal.
Proof. Suppose that dim(A) < dim(B) = dim(G). Then by minimality
of our counterexample we have that A′ = [A, A] is a connected definable
subgroup of A.
Since A′ is characteristic in A, and A is normal in B, we get that A′ is
normal in B. Thus, we can work in B/A′ . We denote by “ ” the quotients
by A′ . Note that A is abelian. Then [A, B] ≤ A = Z(A) and Fact 4.7 gives
that [A, B] is connected and definable. Since A′ ≤ [A, B], we deduce the
definability and connectedness of [A, B], a contradiction.

All in all, we are in the following situation:
G is a simply-connected definable group for which G′ is either non-definable
or definable but non-connected, and such that any proper normal connected
definable subgroup C of G is central in G.
The group is non-solvable, otherwise by [3] we would have that G′ is definable and connected. Since R(G) is a proper connected definable subgroup
of G, we get that R(G) ≤ Z(G) and therefore R(G) = Z(G)o . Then, by
Proposition 4.6 it follows that G′ is definable and connected, a contradiction.

Natural examples of simply-connected o-minimal groups appear in the
literature: the spin groups or the examples in [19, §1] of solvable o-minimal
groups which are not semialgebraic. However, we would like to stress that
simply-connectedness emerges canonically in the context of locally definable
groups. Indeed, every o-minimal group has a (simply-connected) universal
cover which is a locally definable group. Hence, it seems natural to ask:
Question 4.9. Let G be a locally definable group G which is the universal
covering of a connected o-minimal group. Is G′ a compatible subgroup of
G?
Remark 4.10. We would like to finish this section by pointing out that recently we notice that part of the results in [3] can be generalized to an
abstract model-theoretic context. Let G be a group interpretable in a structure M. Henceforth, definability refers to Meq . We suppose that to each
definable set in Cartesian powers of G is attached a dimension in N, denoted
by dim and satisfying the following axioms:
11

(Definability) If f is a definable function between two definable sets A and
B, then for every m in N the set {b ∈ B | dim(f −1 (b)) = m} is a definable
subset of B.
(Additivity) If f is a surjective definable function between two definable sets
A and B, whose fibers have constant dimension m in N, then dim(A) =
dim(B) + m.
(Finite sets) A definable set A is finite if and only if dim(A) = 0.
We also assume that G satisfies the dcc. In particular, G has a smallest
definable subgroup Go of finite index, the intersection of all of them. Then:
Let G be a solvable group equipped with a dimension and with dcc. Let
A and B be two connected definable subgroups of G which normalize each
other. Then the subgroup [A, B] is definable and connected.
Indeed, applying the reductions in the proof of [3, Thm.6.1], i.e. the
claims 1,2 and 3 in the proof of Theorem 4.8 above, it suffices to handle the
following problem: given a solvable group G equipped with a dimension and
with dcc and such that any proper normal connected definable subgroup
is central, its derived subgroup G′ is definable and connected. Now, if G
is abelian then G′ is trivial and we are done. If G is not abelian, then
an argument in [18, Thm.2.12] shows that there exists a proper normal
connected definable subgroup C of G such that G/C is abelian. This ends
the proof since G′ ≤ C ≤ Z(G) and thus G′ is definable by the corresponding
version of Fact 4.7.
We recall the argument in [18, Thm.2.12]. Take C a proper normal connected subgroup C of G of maximal dimension. Suppose that H := G/C
is not abelian. Then we prove that there exists a proper normal definable
e of G such that C is a subgroup of C
e of finite index and G/C̃
subgroup C
is abelian. This yields a contradiction because (G/C)′ would be finite and
therefore by [3, Fact 3.1] we would obtain that G/C is abelian (just consider
for each g ∈ G the action by conjugation of G over the finite set g G ) . Since
H is solvable there exists n ∈ N, n > 1, such that
1 = H (n) < H (n−1) < · · · < H

where H (k) := [H (k−1) ]′ for each k ∈ N. Let m ∈ N, m > 1, be minimal with
e be the normal definable subgroup of G such that C/C
e
H (m) finite. Let C
=
(m−1)
(m)
(m)
e ≃ H/H . Since E := Z(CH (H
H
and consider H1 := G/C
))
1
1
is an infinite abelian normal definable subgroup of H1 , by maximality of
dim(C) we get that E = H1 , so that H1 is abelian, as required.
5. Malcev’s cross-section
As an application of our previous results, we prove an o-minimal version of
Fact 4.1. We need first to study Levi decompositions in the simply-connected
case. The following lemma follows from [7], for the sake of completeness we
provide a proof which becomes somewhat easier in our particular setting.
12

Lemma 5.1. Let G be a simply-connected definable group. Then there exists
a semisimple simply-connected definable subgroup S of G such that G =
R(G)S and R(G) ∩ S = 1.
Proof. It is enough to prove there is a connected semisimple definable subgroup S such that G = R(G)S and R(G) ∩ S is finite. For, in that case the
quotient
G/R(G) = R(G)S/R(G) ≃ S/(R(G) ∩ S)
is simply-connected by Proposition 4.3. Then, by Fact 2.7, the finite normal
subgroup R(G) ∩ S of S must be trivial. In particular, S ≃ G/R(G) is
simply-connected, as required.
Suppose first that G ≤ GL(n, R) is linear. Let g = r + s be a Levi
decomposition of the Lie algebra g of G, where r denotes the radical of
g. We note that Lie(R(G)) = r. Indeed, since R(G) is solvable, its Lie
algebra Lie(R(G)) is solvable [2, Lem.3.7] and therefore Lie(R(G)) ⊆ r.
In particular, since G/R(G) is semisimple, it follows from Fact 2.5 that
g/Lie(R(G)) is semisimple and so Lie(R(G)) = r. On the other hand, since
s = [s, s] = [a(s), a(s)] is algebraic, there is an algebraic group S1 of GL(n, R)
whose Lie algebra is s. Therefore S := S1o is a connected semisimple definable
subgroup of G such that G = R(G)S and R(G) ∩ S is finite, as desired.
Now, suppose that G is almost-linear, i.e. there is a finite normal (central)
subgroup N of G such that G/N is linear. Let π : G → G/N be the canonical
projection, and let R1 := R(G)N/N be the radical of G/N . By the above
there exists a connected semisimple definable subgroup S1 of G/N such that
G/N = R1 S1 and R1 ∩ S1 = 1. Then for the connected semisimple definable
subgroup S := π −1 (S1 )o of G we have that G = R(G)S and R(G) ∩ S is
finite, as required.
For the general case, recall that the kernel of the definable homomorphism
Ad : G → Aut(g) is Z(G), and so the definable group G/Z(G)o is almostlinear and simply-connected by Proposition 4.3. Denote π : G → G/Z(G)o
the canonical projection and let R1 := R(G/Z(G)o ) = R(G)/Z(G)o . Let S1
be a simply-connected semisimple definable subgroup of G/Z(G)o such that
G/Z(G)o = R1 S1 and R1 ∩ S1 = 1.
Consider the connected definable subgroup B := π −1 (S1 ) of G. By Proposition 4.3, since both S1 = B/Z(G)o and Z(G)o are simply-connected, we
have that B is simply-connected. In particular, S := [B, B] is also definable and simply-connected by Theorem 4.8. Let us show that Z(G)o ∩ S
is finite. Indeed, consider an ℵ1 -saturated elementary extension R1 of R,
as well as the realizations G(R1 ), Z(G)o (R1 ) = Z(G(R1 ))o , S1 (R1 ) and
B(R1 ) = π(R1 )−1 (S1 (R1 )) in R1 of the definable groups G, Z(G)o , S1 and
B respectively. To be semisimple and/or simply-connected is preserved under elementary extensions. Therefore, the simply-connected definable group
B(R1 ) is a central extension of the semisimple group S1 (R1 ) and so
Z(G(R1 ))o ∩ [B(R1 ), B(R1 )]n ⊆ Z(B(R1 ))o ∩ [B(R1 ), B(R1 )]n
13

is finite by Fact 4.5. In particular, since [B(R1 ), B(R1 )] is definable by
Proposition 4.3, by saturation there exists n0 ∈ N such that
Z(G(R1 ))o ∩ [B(R1 ), B(R1 )]n0 = Z(G(R1 ))o ∩ [B(R1 ), B(R1 )]n

for all n ≥ n0 . We deduce that

Z(G)o ∩ [B, B]n0 = Z(G)o ∩ [B, B]n

for all n ≥ n0 and therefore Z(G)o ∩ S = Z(G)o ∩ [B, B]n0 is finite.
On the other hand, by Fact 2.5 we have that S1′ = S1 . Thus, π(S) = S1
and
S1 ≃ SZ(G)o /Z(G)o ≃ S/(Z(G)o ∩ S).
Since S1 is simply-connected, it follows from Fact 2.7 that Z(G)o ∩S is trivial.
In particular, S is a simply-connected semisimple definable subgroup of G.
Moreover, we clearly have that G = R(G)S and (R(G) ∩ S)o E R(S) = 1,
as required.

In [20] the authors show that if G is a connected definable group and H
is a contractible normal definable subgroup of G then there is a continuous
section of the projection map G → G/H. The other classical result concerning the existence of cross-sections is, in the o-minimal setting, an easy
consequence of the results in [4]:
Lemma 5.2. Let G be a connected definable group and let H be a normal
connected definable subgroup of G. If G/H is contractible then exists a
continuous definable section σ : G/H → H such that π ◦ σ = id.
Proof. By [4, Cor.2.4] the projection map G → G/H is a definable fibration. Since G/H is contractible, there exists a continuous definable map
F : G/H × [0, 1] → G/H such that F (−, 0) = id and F (−, 1) is the constant
function c : G/H → G/H : ḡ 7→ 1̄. Consider the lifting e
c : G/H → G : ḡ 7→ 1
of c. By the homotopy lifting property of the projection π with respect to
all definable sets, there is a continuous definable map
Fe : G/H × [0, 1] → G

such that π ◦ Fe = F and Fe(−, 1) = e
c. In particular, the continuous definable
e
map σ := F (−, 0) : G/H → G satisfies π ◦ σ = id, as desired.

We already have all the ingredients to prove the existence of cross-sections
in the simply-connected case:

Theorem 5.3. Let G be a simply-connected definable group and let H E
G be a connected definable subgroup. Let π : G → G/H be the natural
homomorphism. Then there exists a continuous definable section σ : G/H →
H such that π ◦ σ = id.
Proof. We prove it by induction on dim(G). The initial case dim(G) = 0
is obvious, so we assume that dim(G) ≥ 1 and the statement holds for
all simply-connected definable groups of dimension less than dim(G). Let
14

H E G be a connected definable subgroup and π : G → G/H the canonical
projection. Note that both H and G/H are simply-connected by Proposition
4.3.
Claim. If there are proper connected definable subgroups A1 and B1 of G/H
such that A1 B1 = G/H and A1 ∩ B1 = 1 then there exists a continuous
definable section σ : G → G/H.
Proof. The map
φ : A1 × B1 → G/H : (a, b) 7→ ab
is clearly a definable homeomorphism. In particular, since π1 (A1 × B1 ) =
π1 (A1 ) × π1 (B1 ) by [10, Lem.2.2] and φ∗ : π1 (G/H) → π1 (A1 × B1 ) is an
isomorphism, it follows that both A1 and B1 are simply-connected. Moreover, by Proposition 4.3 the proper definable subgroups A := π −1 (A1 ) and
B := π −1 (B1 ) of G are simply-connected.
By induction there are continuous definable sections
σA : A1 → A

&

σB : B1 → B

such that π ◦ σA = id and π ◦ σB = id. Consider the continuous definable
maps
σA×B : A1 × B1 → A × B : (x, y) 7→ (σA (x), σB (y))
and
ψ : A × B → G : (x, y) 7→ xy.
Then
σ := ψ ◦ σA×B ◦ φ−1 : G/H → G
is a continuous definable map which satisfies π ◦ σ = id, as desired.

By Proposition 5.1 there exists a definable simply-connected S1 of G/H
such that G/H = R1 S1 and R1 ∩ S1 = 1, where R1 = R(G/H). Thus,
either G/H = R1 or G/H = S1 by the Claim above. If G/H = R1 then
by Corollary 4.4 and Lemma 5.2 there exists a continuous definable section
σ : G/H → G, so we can assume that G/H = S1 is semisimple. Moreover,
again by the Claim and Fact 2.5 we can assume that G/H is definably
simple.
Next, suppose that R(H) is not trivial. Since H is normal in G, and
R(H) is characteristic in H, we get that R(H) is normal in G. Thus, by
Proposition 4.3 the connected definable group G/R(H) is simply-connected.
By induction there is a continuous definable section σ0 : G/H → G/R(H)
of the projection map G/R(H) → G/H. On the other hand, since R(H)
is solvable and simply-connected, by Corollary 4.4 the group R(H) is contractible. Thus, by [20, Thm.5.1] we also have a continuous definable section
σ1 : G/R(H) → G of the projection G → G/R(H). In particular, σ := σ1 ◦σ0
is the desired section of π : G → G/H.
Finally, since we can assume that H is definable simple and G/H is
semisimple, it follows that G is semisimple. In particular, there are definably simple normal definable subgroups C1 , . . . , Cℓ of G such that G ≃
15

C1 × · · · × Cℓ . Since G/H is definably simple, we can assume that π(C1 ) =
G/H. Moreover, ker(π) ∩ C1 is a finite normal subgroup of C1 and therefore
π ↾C1 : C1 → G/H is a definable isomorphism by Fact 2.7. The inverse of
π ↾C1 gives the required continuous definable section σ : G/H → G.

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Departamento de Álgebra; Facultad de Matemáticas; Universidad Complutense de Madrid; Plaza de Ciencias, 3; Ciudad Universitaria; 28040 Madrid;
Spain
E-mail address: eliasbaro@pdi.ucm.es
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