LINEARITY OF GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

OLIVIER FRÉCON

Abstract. We consider an arbitrary o-minimal structure M and a definably connected definable group G. The main theorem provides definable real closed fields R_{1}, …, R_{k} such that G/Z(G) is definably isomorphic to a direct product of definable subgroups of GL_{n_{1}}(R_{1}), …, GL_{n_{k}} (R_{k}), where Z(G) denotes the center of G. It follows from this result a Levi decomposition for G, and that [G, G]Z(G)/Z(G) is definable and definably isomorphic to a direct product of semialgebraic linear groups over R_{1}, …, R_{k}.

1. Introduction

By a result of Otero, Peterzil and Pillay [20], it is well-known that, if a group G is definable in an o-minimal expansion ℛ of a real closed field (R, <, +, · , ⋯), and if G is definably connected, then G/Z(G) is definably isomorphic to a definable subgroup of GL_{n}(R) for an integer n (Fact 5.1).

For groups definable in arbitrary o-minimal structures, a similar result was proven by Peterzil, Pillay and Starchenko [21] for centerless definably connected definable groups: such a group is definably isomorphic to a direct product of definable subgroups of GL_{n_{1}}(R_{1}), … GL_{n_{k}}(R_{k}) for definable real closed fields R_{1}, …, R_{k} and integers n_{1}, …, n_{k} (Fact 2.12).

In this paper, we work inside a fixed arbitrary o-minimal structure M = (M, < , ⋯), and definable means M-definable (with parameters). Our aim is to unify the previous two theorems, which will be used extensively, and as a result, we prove the following.

Main Theorem 5.17. – Let G be a definably connected definable group. Then G/Z(G) is the direct product of definable groups \overline{H_{1}}, …, \overline{H_{k}} such that for every i ∈ {1, …, k} there are a definable real closed field R_{i}, an integer n_{i} and a definable isomorphism between \overline{H_{i}} and a definable subgroup of GL_{n_{i}}(R_{i}).

Moreover, we show that G is the central product of H_{1}, …, H_{k} (Corollary 5.18). The proof of Theorem 5.17 uses numerous results on groups definable in o-

minimal structures. The main new ingredients come from the theory of groups of finite Morley rank, namely pseudo-tori, U_{R}-groups for a definable real closed field R, and U-groups (Definitions 3.1, 3.9 and 3.23). Thanks to these notions, we can provide two results on the structure of definable groups, which are important for the main result.

Date: January 29, 2016.

2010 Mathematics Subject Classification. 03C64.

Key words and phrases. o-minimal structure, levi subgroup, real closed field.

Theorem 3.29. – Any nilpotent definably connected definable group G is the central product of a pseudo-tori by a U-group.

We recall that a group is said to be definably simple if it has no proper non-trivial normal definable subgroup.

Theorem 4.10. – Let G be a definably connected definable group. Then G has a normal U-group U such that G/U is a central extension of a direct product of definably simple definable groups.

We note that, in the Main Theorem, the subgroups \overline{H_{i}} of GL_{n_{i}}(R_{i}) are not necessarily semialgebraic. However, by using the analysis of linear groups in [22], we may obtain a structure result, closer to the semialgebraicity.

Corollary 5.19. – Let G be a definably connected definable group. Then the group G′Z(G)/Z(G) is a direct product of definably connected definable groups \overline{H_{1}}, …, \overline{H_{k}} such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism between \overline{H_{i}} and a semialgebraic linear group over R_{i}.

We show that, thanks to our main result, we may generalize the Levi decomposition, obtained by Conversano and Pillay [6] for groups definable in an o-minimal expansion of a real closed field, to groups definable in an arbitrary o-minimal structure.

For this subject, there is a problem with semisimple groups. Indeed, a semisimple group is defined to be a definably connected definable group with no infinite abelian normal subgroup (Definition 6.1). However, Conversano exhibited a definably connected definable group G with no semisimple subgroup S such that G = RS for a normal solvable subgroup R. In order to remedy to this problem, Conversano and Pillay introduced in [6] ind-definable semisimple subgroups, and they provide the Levi decomposition with these subgroups (Fact 6.3).

In this paper, we introduce quasi-semisimple groups as definably connected definable groups with no decomposition of the form RH for a normal definable solvable subgroup R and a proper definable subgroup H (Definition 6.1). For such a group S, the derived subgroup is perfect and S/Z(S) is semisimple. Then we provide a Levi decomposition for any definably connected definable group (Theorem 6.6 below). Furthermore, we show that if G is any definably connected group definable in an o-minimal expansion of a real closed field, its maximal ind-definable semisimple subgroups are precisely the derived subgroups of its maximal quasi-semisimple subgroups (Corollary 6.7).

Theorem 6.6. – Let G be a definably connected definable group. Then G has a maximal quasi-semisimple subgroup S, unique up to conjugacy in G. Moreover, there is a normal solvable definable subgroup R such that G = RS and G ∩ S ≤ S.

The organization of this paper is as follows. In §2, we recall known results and give some useful corollaries. The purpose of §3 is the analysis of nilpotent groups (Theorem 3.29). In particular, we introduce pseudo-tori, U_{R}-groups and U-groups, which are fundamental notions for this paper. In §4, we study the group actions on a solvable group, and then we obtain a structure theorem for any definably connected definable group (Theorem 4.10). In §5, we prove the main result of this paper (Theorem 5.17). In §6, we apply the main result to Levi decomposition (Theorem 6.6).

2. Preliminaries

The basic reference for o-minimal structures is [27] (see [19] for a survey on groups definable in an o-minimal structure).

By [12], in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. Actually, any group definable in an o-minimal structure eliminates imaginaries. More precisely, the following result is due to M. Edmundo.

Fact 2.1.. – [11, Theorem 7.2] Let G be a definable group, and let {T (x) : x ∈ X} be a definable family of non-empty definable subsets of G. Then there is a definable function t : X → G such that for all x, y ∈ X we have t(x) ∈ T (x) and if T (x) = T (y) then t(x) = t(y).

2.1. Nilpotent definable groups. We recall two general facts on nilpotent groups definable in an o-minimal structure, and more generally. Any group definable in an o-minimal structure 𝒩 satisfies the descending chain condition on 𝒩 -definable subgroups [25, Remark 2.13 (ii)]. In particular, it is an M_{c}-group, that is a group with descending chain condition on centralizers. Thus, the following two facts are satisfied by nilpotent groups definable in an o-minimal structure.

For every group G, we denote by Z_{0}(G) = 1 the trivial group, and we define Z_{i}(G) for each integer i by Z_{i+1}(G)/Z_{i}(G) = Z(G/Z_{i}(G)).

Fact 2.2.. – [13, Lemma 3.7.10] Any infinite nilpotent M_{c}-group has infinite center.

More generally, if a group G has a finite subset X such that Z(G) = C_{G}(X), and if H is a normal subgroup such that H ∩ Z_{k}(G) is infinite for an integer k, then Z(G) ∩ H is infinite.

Proof – There is a smallest integer j such that B = Z_{j+1}(G) ∩ H is infinite. Then [g, B] is contained the finite subgroup Z_{j}(G) ∩ H for each g ∈ G, and the index of C_{B}(g) in G is finite. Thus B/C_{B}(X) is finite, and Z(G) ∩ B has finite index in B, so Z(G) ∩ H = Z(G) ∩ B is infinite. □

Fact 2.3.. – (see [2, Lemma 6.3] for a special case) Let G be a nilpotent group. We consider an expansion 𝒢 = (G, · , ⋯) of the group G such that G has the descending chain condition on its 𝒢 -definable subgroups. If H is a 𝒢 -definable subgroup of G of infinite index, the index of H in N_{G}(H) is infinite too.

Proof – We consider the normal subgroup K = ∩_{g∈G}H^{g} of G and, for each g ∈ G, the subgroup C_{g} = {x ∈ G | [g, x] ∈ K}. By descending chain condition on 𝒢 -definable subgroups, K and Z = ∩_{g∈G}C_{g} are 𝒢 -definable, and there is a finite subset X of G such that Z = ∩_{g∈X}C_{g}. We note that Z/K = Z(G/K) is the center of G/K, so by Fact 2.2, it is infinite. Since it normalizes H/K, the subgroup Z normalizes H. Moreover, since Z ∩ H ≤ Z contains K, the subgroup Z ∩ H is normal in G, and since it is contained in H, we have Z ∩ H = K. Thus the index of H in N_{G}(H) ≥ Z is infinite. □

2.2. Connected component. For every definable group G, we denote by G^{∘} the definably connected component (of the identity) in G. It is the smallest definable subgroup of G of finite index in G [25, Proposition 2.12]. A group G is said to be definably connected if G = G^{∘}.

In this section, we show that for any definable group G, this subgroup G^{∘} is definable in the pure group (G, ·), and has no proper subgroup of finite index: every subgroup of finite index is definable and G^{∘} is the smallest subgroup of finite index (Proposition 2.9 below). In particular, the definably connected component of a definable group is independant from the language.

The proof of Proposition 2.9 requires several facts.

It follows from [25] that any definable group G has largest definably connected definable solvable normal subgroup R(G), called the solvable radical of G. However, another definition of solvable radical is used in [1].

Fact 2.4.. – [1, Lemma 4.5] Let G be a definable group. The subgroup generated by all normal solvable subgroups of G is definable and solvable.

Moreover, by [7], any definable group G has a largest nilpotent normal subgroup F (G), and this subgroup is definable by [11, Lemma 6.7].

For each group G, we denote by G′ = [G, G] the derived subgroup. We recall that, for a definable group G, this subgroup is not necessarily definable (Conversano exhibits a counter-example [5, Example 3.1.7]). However, Baro, Jaligot and Otero [1] show its definability for a large class of definable groups.

The derived subgroup of a solvable definably connected definable group has been studied in [11, Theorem 6.9], and a precision is given in [1, Proposition 5.5].

Fact 2.5.. – Let G be a solvable definably connected definable group. Then the following two conditions are satisfied:

• [11, Theorem 6.9] its derived subgroup G′ is contained in F (G);

• [1, Proposition 5.5] the group G/F (G)^{∘} is abelian and divisible.

Fact 2.6 describes the structure of nilpotent groups, where a group G is the central product of two subgroups H and K if G = HK and [h, k] = 1 for each (h, k) ∈ H × K. We denote this by G = H ∗ K.

Fact 2.6.. – Let G be a nilpotent definable group.

(1) [11, Theorem 6.10] and [1, Lemma 3.10 (c)] G^{∘} is divisible and G has a finite characteristic subgroup F such that G = G^{∘} ∗ F.

(2) [28, Theorem 4.12 (Chernikov)] and [8, Corollary 1.5.12] The torsion subgroup of G^{∘} is central in G.

Corollary 2.7.. – Let G be a nilpotent definably connected definable group. Then any definable subgroup of G containing Z(G) is definably connected.

Proof – Since Z(G) contains the torsion part of G by Fact 2.6 (2), the group G/Z(G) is torsion-free. In particular, each definable subgroup of G/Z(G) is definably connected. But G is divisible by Fact 2.6 (1), so the torsion part of Z(G) is divisible, and Fact 2.6 (1) applied with Z(G) shows that Z(G) is definably connected. Hence, for any definable subgroup H/Z(G) of G/Z(G), the subgroup Z(G) is contained in H^{∘}, and we have H = H^{∘}Z(G) = H^{∘}. □

The following result is a very important theorem for groups definable in an ominimal structure. It is used in the proof of Proposition 2.9 below.

Fact 2.8.. – [21, Theorem 4.1] Let G be an infinite (G, ·)-definably connected definable group. Assume G has no nontrivial abelian normal subgroup. Then G is the direct product of (G, ·)-definable subgroups H_{1}, …, H_{k} such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism between H_{i} and a semialgebraic subgroup of GL_{n_{i}}(R_{i}). Moreover, H_{i} is (H_{i}, ·)- definably simple and H_{i}^{∘} is definably simple.

Proposition 2.9.. – Let G be definable group. Then G^{∘} is definable in the pure group (G, ·). Moreover, G^{∘} has no proper subgroup of finite index.

Proof – Every (G, ·)-definable subgroup of G is definable, so G has a smallest (G, ·)-definable subgroup of finite index. In particular, this subgroup is definable, contains G^{∘} as a subgroup of finite index, and it has no proper (G, ·)-definable subgroup of finite index. So we may assume that G is (G, ·)-definably connected, and we have just to prove that G has no proper subgroup of finite index.

Let N be a subgroup of G of finite index n. We show that N = G. We may assume that N is contained in G^{∘}. Moreover, since the index of N in G is finite, N contains a G-normal subgroup of finite index, and we may assume that N is normal in G. Let X = {x^{n} | x ∈ G}. In particular, X is a definable subset of N.

We show that XX contains R(G). By Fact 2.6 (1), the definable subgroup F (G)^{∘} is divisible, so it is contained in X. By Fact 2.5, the quotient group R(G)/F (G)^{∘} is divisible, so XF (G)^{∘} ⊆ XX contains R(G).

Let Z be the subgroup of G^{∘} generated by all its normal solvable subgroups. It is definable and solvable (Fact 2.4), so Z/R(G) is finite. In particular, G^{∘}/Z has no non-trivial abelian normal subgroup, and Fact 2.8 implies that G^{∘}/Z is the direct product of definable subgroups H_{1}/Z, …, H_{k}/Z such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism f_{i} between H_{i}/Z and a semialgebraic subgroup L_{i} of GL_{n_{i}}(R_{i}), and such that H_{i}^{∘}Z/Z is definably simple. Moreover, since G^{∘}/Z is definably connected, H_{i}/Z is definably connected for each i, so L_{i} ≃ H_{i}/Z is definably simple.

We show that N contains a (G, ·)-definable set containing H_{i} ∩N for each i. Let i ∈ {1, …, k}. Since L_{i} is a definably simple semialgebraic subgroup of GL_{n_{i}}(R_{i}), each element g ∈ L_{i} is a product of a semisimple element s ∈ L_{i} and a unipotent element u ∈ L_{i}. Then s is contained in a maximal torus T_{i} of L_{i} and u is contained in a unipotent subgroup U_{i} of L_{i}. In particular, T_{i} and U_{i} are divisible, so f_{i}^{−1}(s) and f_{i}^{−1}(u) are contained in XZ/Z, and f_{i}^{−1}(g) belongs to XXZ/Z. Consequently H_{i} is contained in XXZ. Since XX ⊆ N contains R(G), there is a finite subset W of Z ∩ N such that Z ∩ N = W R(G). Therefore H_{i} ∩ N is contained in

XXZ ∩ N = XX(Z ∩ N) = XXW R(G) ⊆ XXW XX

Thus M = (H_{1} ∩ N) ⋯ (H_{k} ∩ N) is a subgroup of finite index in N contained in XXW XX. We consider a finite subset E of N such that N = EM, and we obtain N = EXXW XX. So N is (G, ·)-definable, and N = G. □

2.3. Definable fields. The following fundamental results are due to Pillay, and Peterzil and Steinhorn respectively. They are crucial for us.

Fact 2.10.. – [25, Theorem 3.9 and Proposition 3.11] Let K be an infinite definable field. Then K is real closed or algebraically closed. It is real closed if and only if its dimension is 1.

Fact 2.11.. – [24, Theorem 4.1] Let 𝒦 = (K, +, 0, ·) be an infinite definable ring without zero divisors. Then K is a division ring and there is a one-dimensional definable subring R of K which is a real closed field such that K is either R, R(\sqrt{−1}), or the ring of quaternions over R.

Lemma 2.14 and Proposition 2.15 are very useful for this paper. The proof of Proposition 2.15 is based on the following very important fact (Fact 2.12), and on the study of abelian definable subgroups of the general linear group over a definable real closed field (Fact 2.13). Moreover, we note that the proof of Fact 2.12 is based on the theory of nonorthogonality from [21].

Fact 2.12.. – [21, Theorems 3.1 and 3.2] Let G be a definably connected definable centerless group. Then G is definably isomorphic to a direct product H_{1} × ⋯ × H_{k}, where, for each i = 1, …, k, there is a definable real closed field R_{i} such that H_{i} is a definable subgroup of GL_{n_{i}}(R_{i}) for each i = 1, …, k.

Fact 2.13.. – Let G be a commutative definably connected definable subgroup of GL_{n}(R) where R is a definable real closed field. Then the following three conditions hold:

(1) [22, Fact 3.1] if G is semialgebraic, it is semialgebraically isomorphic to a group of the form SO_{2}(R)^{m} × (R_{>0}^{∗})^{l} × (R_{+})^{k}; [22, Lemma 3.9] every definably connected definable subgroup H of G has a definable complement in G;

(2) [22, Special case of Proposition 3.10] G is definably isomorphic to a linear semialgebraic group over R.

Lemma 2.14.. – Let R and S be two definable real closed fields. If the groups R_{+} and S_{+} are definably isomorphic, then the fields R and S are definably isomorphic.

Proof – Let f : R_{+} → S_{+} be a definable isomorphism. In particular f(1) is non-zero and we may consider the map g : R → S defined by g(x) = f(x)f(1)^{−1}. Then g is a definable isomorphism from R_{+} and S_{+} such that g(1) = 1.

Now, for each α ∈ R, the subset A_{α} = {x ∈ R | g(xα) = g(x)g(α)} of R is a definable subgroup of R_{+} containing 1. So we obtain A_{α} = R for each α ∈ R and g is a field isomorphism. □

Proposition 2.15.. – Let ℛ = (R, +, ·) and 𝒮 = (S, ⊕, ∗) be two definable real closed fields. If there is an infinite definable ℛ-linear group H definably isomorphic to an 𝒮 -linear group, then the fields ℛ and 𝒮 are definably isomorphic.

Proof – We may assume that H has no proper infinite definable subgroup. Then H is definably isomorphic either to SO_{2}(R) or to R_{>0}^{∗} or to R_{+} (Fact 2.13 (1) and (3)). By the same way and by Lemma 2.14, we may assume that H is definably isomorphic either to SO_{2}(S) or to S_{>0}^{∗}.

If H is definably isomorphic to SO_{2}(R), then it has torsion and it is definably isomorphic to SO_{2}(S). We consider the semi-direct product G = (R_{+}^{2} × S_{+}^{2}) ⋊ H where H ≃ SO_{2}(R) acts ℛ-linearly on R_{+}^{2} and such that H ≃ SO_{2}(S) acts 𝒮 - linearly on S_{+}^{2}. In particular, G is centerless and it has no decomposition H = A × B as a direct product of two proper subgroups. By Fact 2.12, there is a definable real closed field 𝒯 = (T, ⋯) and a definably linear group K ≤ GL_{n}(T) definably isomorphic to G. But K ≃ G is definably connected and 2-solvable, so its derived subgroup K′ ≃ G′ is definably isomorphic to T_{+}^{m} for an integer m and to G′ ≃ R_{+}^{2} × S_{+}^{2}. This implies that the groups R_{+}, S_{+} and T_{+} are definably isomorphic, so the fields ℛ and 𝒮 are definably isomorphic by Lemma 2.14.

Hence we may assume that H is torsion-free, so it is definably isomorphic either to R_{>0}^{∗} or to R_{+}, and to S_{>0}^{∗}. If it is definably isomorphic to R_{>0}^{∗}, we consider the semi-direct product G = (R_{+} ×S_{+})⋊H where H ≃ R_{>0}^{∗} acts ℛ-linearly on R_{+} and such that H ≃ S_{>0}^{∗} acts 𝒮 -linearly on S_{+}. As in the previous paragraph, Fact 2.12 provides a real closed field 𝒯 = (T, ⋯) and a definably linear group K ≤ GL_{n}(T) definably isomorphic to G, and we conclude that the groups R_{+}, S_{+} and T_{+} are definably isomorphic, so the fields ℛ and 𝒮 are definably isomorphic by Lemma 2.14.

Thus we may assume that H, R_{+}, and S_{>0}^{∗} are definably isomorphic. The group H × R_{>0}^{∗} ≃ R_{+} × R_{>0}^{∗} acts ℛ-linearly on R_{+} × R_{+} where the action is defined by (a, t) · (x, y) = (tx, atx + ty). We consider the semi-direct product G = ((R_{+} × R_{+}) × S_{+}) ⋊ (H × R_{>0}^{∗}) where H × R_{>0}^{∗} acts as above on R_{+} × R_{+}, the group R_{>0}^{∗} acts trivially on S_{+}, and H ≃ S_{>0}^{∗} acts 𝒮 -linearly on S_{+}. Again G is centerless and has no decomposition as a direct product of two proper subgroups, so Fact 2.12 provides a real closed field 𝒯 = (T, ⋯) and a definably linear group K ≤ GL_{n}(T) definably isomorphic to G. As above we conclude that the groups R_{+}, S_{+} and T_{+} are definably isomorphic, so the fields ℛ and 𝒮 are definably isomorphic by Lemma 2.14. □

2.4. The structure of solvable groups by Edmundo. Edmundo gives in [11] a precise description of the structure of solvable groups. His main result, namely Fact 2.24, is very useful for a key result of the analysis of nilpotent groups (Proposition 3.22). Before stating it, we specify the terminology.

In [24], Peterzil and Steinhorn introduced the notion of definable compactness in o-minimal structures.

Definition 2.16.. – Let G be a definable group. We say that G is definably compact if for every definable continuous embedding σ : (a, b) ⊆ M → G, where −∞ ≤ a < b ≤ +∞, there are c, d ∈ G such that lim_{x→a^{+}} σ(x) = c and lim_{x→b^{−}} σ(x) = d, where the limits are taken with respect to the topology on G.

We recall that a semisimple group is defined to be a definably connected definable group with no infinite abelian normal subgroup (Definition 6.1).

Fact 2.17.. – [11, Corollary 4.8] (see also [23, Corollary 5.4]) Let G be a definably connected definably compact definable group. Then G is either abelian or G/Z(G) is definably semisimple definable group. In particular, if G is abelian then it is abelian.

Fact 2.18.. – [11, Lemma 3.14] Let A be a normal definable subgroup of a definable group U. Then U is definably compact if and only if A and U/A are definably compact.

Fact 2.19.. – [24, Proof of Theorem 4.1] (see also Fact 2.11) Let 𝒦 = (K, +, 0, ·) be an infinite definable ring without zero divisors. Then K is not definably compact.

Miller and Starchenko introduced linearly bounded groups in [18], and Edmundo has investigated semi-bounded groups in [10] (see also the introduction of [11]), a special case of linearly bounded groups. We do not give the definitions, just their main properties.

First at all, we recall that any o-minimal ordered groups is abelian by [26, Theorem 2.1].

Fact 2.20.. – [26, Theorem 2.1 and Lemma 2.2] Let (G, <, ·) be an o-minimal ordered group. Then the group G is abelian and divisible, and it has no non-trivial proper definable subgroup.

Fact 2.21.. – [18, Theorem A (Growth Dichotomy)] Let ℛ be an o-minimal expansion of an ordered group (R, <, +). Then exactly one of the following holds:

• ℛ is linearly bounded;

• ℛ defines a binary operation · such that (R, <, +, ·) is an ordered real closed field.

Corollary 2.22.. – If ℛ = (R, <, +, ·) is a definable real closed field, then the ordered group (R_{>0}, <, ·) is either definably isomorphic to (R, <, +) or it is linearly bounded.

Proof – If (R_{>0}, <, ·) is not linearly bounded, then by Fact 2.21, there is a binary operation ∗ such that 𝒮 = (R_{>0}, <, · , ∗) is an ordered real closed field. But its additive group (R_{>0}, <, ·) is an ℛ-linear group, so the fields ℛ and 𝒮 are definably isomorphic by Proposition 2.15, hence the groups (R_{>0}, <, ·) and (R, <, +) are definably isomorphic. □

Fact 2.23.. – [10, Special case of Fact 1.6] For an o-minimal expansion ℛ = (R, 0, 1, +, <, ⋯) of an ordered group the following are equivalent:

(1) ℛ is semi-bounded.

(2) There is no ℛ-definable real closed field with domain R.

Moreover, in the following result, two definable o-minimal structures ℐ = (I, < , ⋯) and 𝒥 = (J, <, ⋯) are said to be globally orthogonal if there is no definable bijection between I and J.

Fact 2.24.. – [11, Theorems 5.8 and 5.10] Suppose that U is a definably-connected definable solvable group. Then U has a definable normal subgroup V such that U/V is a definably compact definable solvable group and V = K × W_{1} × ⋯ × W_{s} × V_{1} × ⋯ × V_{k}. Here K is the definably-connected, definably compact normal subgroup of U of maximal dimension. For each j ∈ {1, …, s} (resp. i ∈ {1, …, k}), there is a semi-bounded definable o-minimal expansion 𝒥_{j} of a group (resp., a definable o-minimal expansion ℐ_{i} of a real closed field) all of which are pairwise globally orthogonal such that W_{j} is a direct product of copies of the additive group of 𝒥_{j} and V_{i} is definably isomorphic to an ℐ_{i}-definable solvable group.

Moreover, for each i ∈ {1, …, k}, we have V_{i} = W × V where W is the maximal ℐ_{i}-definable subgroup of V_{i} which is a direct product of copies of the linearly bounded one-dimensional torsion-free ℐ_{i}-definable group. The group V is an ℐ_{i}-definable group such that Z(V) has an ℐ_{i}-definable subgroup Z such that Z(V)/Z is a direct product of copies of the linearly bounded one-dimensional torsion-free ℐ_{i}-definable group. There are ℐ -definable subgroups 1 < Z_{1} < ⋯ < Z_{m} = Z such that, for each l ∈ {1, … m}, the group Z_{l}/Z_{l−1} is the additive group of ℐ_{i}, and there is an ℐ_{i}-definable embedding of V/Z(V) into GL_{n}(I).

3. Nilpotent groups

The structure of solvable groups by Edmundo [11] (see §2.4) provides valuable information on nilpotent groups definable in an o-minimal structure. By using these results together with methods from groups of finite Morley rank, we obtain a new decomposition of nilpotent groups based on pseudo-tori and U_{R}-groups (Definitions 3.1 and 3.9, Theorem 3.29).

The structure of nilpotent groups in this new language is very effective for the study of group actions in §4.

3.1. pseudo-tori. Cherlin defined in [4] a good torus and a decent torus as analogues of an algebraic torus for groups of finite Morley rank. These groups are defined from the torsion, and a more general notion of a torus was introduced in [17]: the pseudo-tori whose definition for the finite Morley rank context is very close to the following definition.

Definition 3.1.. – A pseudo-torus is a definably connected nilpotent definable group T such that no definable quotient group T/N is definably isomorphic to the additive group R_{+} of a definable real closed field R.

Remark 3.2.. – Any definable quotient T/N of a pseudo-torus T is a pseudo-torus.

The following result gives examples of pseudo-tori, which encompasse definably compact groups, linearly bounded groups and semi-bounded groups (see §2.4). It will be useful for the proof of Proposition 3.22.

Lemma 3.3.. – Let G be a solvable definably connected definable group. Suppose that G satisfies one of the following three conditions:

• G is definably compact;

• there is a definable expansion ℛ of the group G such that ℛ is a linearly bounded structure;

• there is a definable expansion ℛ of the group G such that ℛ is a semi-

bounded structure. Then G is a pseudo-torus.

Proof – If G is definably compact, then it is abelian by Fact 2.17. Now, since any definable quotient of G is definably compact (Fact 2.18), it follows from Fact 2.19 that G is a pseudo-torus.

If there is a definable expansion ℛ = (G, <, +, ⋯) of the group G such that ℛ is a linearly bounded structure or a semi-bounded structure, then G is abelian and has no non-trivial proper definable subgroup (Fact 2.20). Since it follows from Facts 2.21 and 2.23 that there is no definable binary operation · such that (G, <, +, ·) is a real closed field, G is a pseudo-torus. □

We start our study of pseudo-tori. The following result is used in the proof of Lemma 3.5.

Fact 3.4.. – [11, Corollary 7.3 (1)] (see also [23, Theorem 1.1] and Fact 2.11) Let A and B be two definable abelian groups. If there is an infinite definable family of definable homomorphisms from A into B, then there is a definable real closed field whose additive group is definably isomorphic to a definable subgroup of B and a quotient of definable subgroups of A.

Lemma 3.5.. – Let T be a pseudo-torus and B be a nilpotent definable group. Then any definable family 𝒜 of homomorphisms from T to B is finite.

Proof – We proceed by induction on the dimension of B. Since for each α ∈ 𝒜, the image Im α ≃ T/Ker α of α is definably connected, we have Im α ≤ B^{∘} and we may assume that B is definably connected. We assume toward a contradiction that 𝒜 is infinite. In particular, B is infinite.

We assume toward a contradiction that B has a proper infinite normal definable subgroup A. For each α ∈ 𝒜, we consider \overline{α} : T → B/A defined by \overline{α}(t) = α(t)A. Then the definable family \{\overline{α} | α ∈ 𝒜 \} is finite by induction hypothesis, and there exists α ∈ 𝒜 such that the definable family ℬ = \{β ∈ 𝒜 | \overline{β} = \overline{α}\} is infinite. For each β ∈ ℬ, the map u_{β} : T → A defined by u_{β}(t) = β(t)α(t)^{−1} is a definable group homomorphism, and since ℬ is infinite, the definable family {u_{β} | β ∈ ℬ} is infinite too, contradicting our induction hypothesis. Hence B has no proper infinite normal definable subgroup. In particular B is abelian (Fact 2.2), so it has no proper infinite definable subgroup.

Let K be the intersection of the subgroups Ker α for α ∈ 𝒜. Since T/Ker α ≃ Im α ≤ B is abelian for each non-zero element α ∈ 𝒜, the quotient group T/K is abelian. For each α ∈ 𝒜, we consider α̃ : T/K → B defined by α̃(tK) = α(t). Since 𝒜 is infinite, \overline{𝒜} = \{α̃ | α ∈ 𝒜 \} is infinite too. Then by Fact 3.4 there is a definable subgroup B_{0} of B such that B_{0} is definably isomorphic to the additive group R_{+} of a definable real closed field R. In particular, B_{0} is infinite and we obtain B = B_{0} by the previous paragraph. But \overline{𝒜} is infinite, so there is a non-zero element α̃ ∈ \overline{𝒜} , and its image Im α̃ = Im α ≃ T/Ker α is definably connected. Hence Im α is an infinite definable subgroup of B and α is a surjective homomorphism by the previous paragraph. Thus we have T/Ker α ≃ Im α = B ≃ R_{+}, contradicting that T is a pseudo-torus and that R is real closed, so 𝒜 is finite. □

Corollary 3.6.. – Let T be a pseudo-torus and G be a definably connected definable group acting definably on T. Then G centralizes T. In particular, T is abelian.

Proof – By Lemma 3.5, the quotient group G/C_{G}(T) is finite, and since G is definably connected, G centralizes T. In particular, the case where G = T acts by conjugation on T shows that T is abelian. □

Proposition 3.7.. – Any nilpotent definable group G has a unique maximal pseudotorus T (G). In particular, any pseudo-torus of G is central in G.

Proof – We proceed by induction on the dimension of G. We may assume that G is definably connected. Let S and T be two maximal pseudo-tori of G.

We show that S and T are central in G. If N_{G}(T) < G, we have T = T (N_{G}(T)) by induction hypothesis, therefore T is a definably characteristic subgroup of N_{G}(T) and we obtain N_{G}(N_{G}(T)) = N_{G}(T). But G is nilpotent, hence we have N_{G}(T) = G contradicting N_{G}(T) < G. This proves that T is normal in G, and T is central in G by Corollary 3.6. In the same way, S is central in G.

We assume toward a contradiction that ST is not a pseudo-torus. Then ST has a definable subgroup N such that ST/N is definably isomorphic to the additive group R_{+} of a definable real closed field R. In particular, the quotient group T S/N is torsion-free and it has dimension one by Fact 2.10. If T is not contained in N, we have T S = T N and T/(T ∩ N) is definably isomorphic to T S/N ≃ R_{+}, contradicting that T is a pseudo-torus. Thus T is contained in N, and in the same way, S is contained in N, contradicting N < ST. This proves that ST is a pseudo-torus, and that T (G) = S = T is well defined. □

Proposition 3.8.. – Let G be a nilpotent definable group, and N be a normal definable subgroup of G. Then T (G/N) = T (G)N/N.

Proof – We proceed by induction on the dimension of G. Since a definable quotient of a pseudo-torus is a pseudo-torus, T (G/N) contains T (G)N/N and we may assume that G/N = T (G/N) is a pseudo-torus.

Let T be a minimal definable subgroup of G among the ones satisfying G = T N. Since G/N is a pseudo-torus, G/N is definably connected and we have G = T^{∘}N. Thus T is definably connected by the minimality of T.

We assume toward a contradiction that T is not a pseudo-torus. Then T has a definable quotient T/M definably isomorphic to R_{+} for a real closed field R. In particular, T/M is torsion-free and has dimension 1 by Fact 2.10. Since (T ∩ MN)/M is a definable subgroup of T/M, it is either equal to T/M or trivial, so we have either T ∩ MN = M or T ≤ MN. In the first case we have

(G/N)/(MN/N) ≃ G/MN = T N/MN ≃ T/M ≃ R_{+}

contradicting that G/N is a pseudo-torus. In the second case we have G = T N = MN, contradicting the minimality of T. Hence T is a pseudo-torus, and we obtain T ≤ T (G) and G = T (G)N. □

3.2. U_{R}-groups. Burdges introduced U_{0,r}-groups in [3] as a concept of unipotence for groups of finite Morley rank. This notion is very effective for the study of groups of finite Morley rank. Another analogue of unipotent algebraic groups, namely the homogeneous U_{0,r}-groups, was proposed in [14] in order to remedy to a weakness of U_{0,r}-groups, since they are not necessarily preserved by passage to definable subgroups. Later, a more precise unipotence notion was introduced in [15, §3.2], very close to Definitions 3.9 and 3.15. This last notion, together with pseudo-tori and the homogeneity of [14], is a crucial tool for some analysis as [15].

We note that N is a normal subgroup of G in the following definition (Fact 2.3).

Definition 3.9.. – Let R be a definable real closed field. A U_{R}-group is a nilpotent definable group G such that, for every maximal proper definably connected definable subgroup N, the quotient group G/N is definably isomorphic to R_{+}.

Remark 3.10.. – Any U_{R}-group is definably connected.

Proposition 3.11.. – Let R be a definable real closed field, and G be a nilpotent definable group. Then any family of U_{R}-subgroups of G generates a U_{R}-subgroup.

In particular, G has a unique maximal U_{R}-subgroup.

Proof – We note that we do not know if any subgroup of G generated by U_{R}- subgroups is definable. We have just to show that any non-necessarily definable subgroup H of G contains a unique maximal U_{R}-subgroup. We proceed by induction on the dimension of G. Since any U_{R}-group is definably connected, we may assume that G is definably connected. Let U and V be two maximal U_{R}-subgroups of H.

We show that U is normal in H. We may assume that U is not normal in G. By induction hypothesis, U is the unique maximal U_{R}-subgroup of N_{H}(U) ≤ N_{G}(U) < G, therefore U is normal in N_{H}(N_{H}(U)) and we obtain N_{H}(N_{H}(U)) = N_{H}(U). But H is nilpotent, hence N_{H}(U) = H and U is normal in H. In particular, UV is a definably connected definable subgroup of H.

We may assume that UV is infinite, therefore UV has a maximal proper definably connected definable subgroup N. If UN = UV, we have U/(U ∩ N) ≃ UV/N and by the maximality of N in UV, the group (U ∩ N)^{∘} is a maximal proper definably connected definable subgroup of U. Since U is a U_{R}-group, the group U/(U ∩ N)^{∘} ≃ R_{+} is torsion-free and U ∩ N is definably connected, so we obtain UV/N ≃ U/(U ∩ N) ≃ R_{+}. In the same way, if V N = UV the groups UV/N and R_{+} are definably isomorphic. But N is proper in UV, so we have either U ≰ N or V ≰ N, and by the maximality of N we have either UN = UV or V N = UV. Hence UV/N is definably isomorphic to R_{+}, and UV is a U_{R}-group. Now by the maximality of U and V, we obtain UV = U = V = U_{R}(H), as desired. □

Thus we may define a radical U_{R}( ·) for each definable real closed field R.

Definition 3.12.. – Let R be a definable real closed field. For each definable group G, we denote by U_{R}(G) the unique maximal U_{R}-subgroup of F (G).

Lemma 3.13.. – Let G be a definable group with a normal definable subgroup N such that G/N is definably isomorphic to R_{+} for a real closed field R. Then N ∩G^{∘} is definably connected and G = UN for an abelian U_{R}-subgroup U.

Proof – Since G/N ≃ R_{+} is torsion-free, it is definably connected and G/N is definably isomorphic to G^{∘}/(N ∩ G^{∘}). Therefore the torsion part of G^{∘}/N^{∘} is (N ∩ G^{∘})/N^{∘}, so it is finite, and Fact 2.6 (1) gives N ∩ G^{∘} = N^{∘}.

Let U be a minimal definable subgroup of G among the ones satisfying U ≰ N. For any u ∈ U \ N the subgroup Z(C_{U}(u)) is definable, abelian and contains u, so Z(C_{U}(u)) = U by minimality of U, and U is abelian. Since G/N ≃ R_{+} is torsion-free, and since its dimension is 1 (Fact 2.10), we have G = UN.

We show that U is a U_{R}-group. Since G/N ≃ R_{+} is definably connected, we have G = U^{∘}N, and U is definably connected by minimality of U. Now the first paragraph applied with U and U/(U ∩ N) ≃ R_{+} shows that U ∩ N is definably connected. But, again by the minimality of U, each proper definable subgroup of U is contained N. Hence U ∩ N is the unique maximal proper definably connected definable subgroup of U. Thus U is a U_{R}-group. □

Proposition 3.14.. – Let R be a definable real closed field, G be a nilpotent definable group, and N be a normal definable subgroup of G. Then

U_{R}(G/N) = U_{R}(G)N/N

Proof – We show that U_{R}(G/N) contains U_{R}(G)N/N. Let M/N be a maximal proper definably connected definable subgroup of U_{R}(G)N/N. Then U_{R}(G)N/M ≃ U_{R}(G)/(U_{R}(G) ∩ M) has no non-trivial proper definably connected definable subgroup, and (U_{R}(G) ∩ M)^{∘} is a maximal proper definably connected definable subgroup of U_{R}(G). Thus U_{R}(G)/(U_{R}(G) ∩ M)^{∘} is definably isomorphic to R_{+} and, by Lemma 3.13, the subgroup U_{R}(G) ∩ M is definably connected. Therefore U_{R}(G)N/M ≃ U_{R}(G)/(U_{R}(G) ∩ M) is definably isomorphic to R_{+}, so U_{R}(G)N/N is a U_{R}-group and it is contained in U_{R}(G/N).

We show that U_{R}(G/N) = U_{R}(G)N/N. We denote by U the preimage in G of U_{R}(G/N). For each maximal proper definably connected definable subgroup M/N of U_{R}(G/N), the group U/M is definably isomorphic to R_{+}, so Lemma 3.13 gives U = U_{R}(G)M. Consequently U_{R}(G)N/N is contained in no proper definably connected definable subgroup of U_{R}(G/N). Since U_{R}(G)N/N is definably connected, we obtain U_{R}(G/N) = U_{R}(G)N/N, as desired. □

3.3. Homogeneous U_{R}-groups. Similarly to the groups of finite Morley rank, we define an homogeneous U_{R}-group [14]. The purpose of this section is to show that any U_{R}-group is homogeneous (Proposition 3.22).

Definition 3.15.. – Let R be a definable real closed field. A U_{R}-group is said to be homogeneous if its definable subgroups are U_{R}-groups.

Remark 3.16.. – Let R be a definable real closed field.

• Any homogeneous U_{R}-group is definably connected and torsion-free.

• Every definable subgroup of a homogeneous U_{R}-group is a homogeneous U_{R}-group.

• By Proposition 3.14, every definable quotient group of a homogeneous U_{R}- group is a homogeneous U_{R}-group.

Lemma 3.17.. – Let R be a definable real closed field. If a nilpotent definable group G has a normal homogeneous U_{R}-subgroup U such that G/U is a homogeneous U_{R}- group, then G is a homogeneous U_{R}-group.

Proof – Let H be a definable subgroup of G. Since G/U is a homogeneous U_{R}- group, HU/U is a U_{R}-group, and by Proposition 3.14, we have H = U_{R}(H)(H ∩U). But U is a homogeneous U_{R}-group, hence H ∩ U and H are U_{R}-groups. □

For the proof of Lemma 3.20, we need G-minimal subgroups.

Definition 3.18.. – Let G be a definable group. A subgroup of G is said to be G-minimal if it is definable, infinite, normal, and minimal for these conditions.

Remark 3.19.. –

• By the descending chain condition on definable subgroups of G [25, Remark 2.13 (ii)], any infinite normal definable subgroup H of a definable group G contains a G-minimal subgroup.

• In a definable group G, every G-minimal subgroup is definably connected.

Lemma 3.20.. – Let R and S be two definable real closed fields, and let G be a nilpotent definable group. If R and S are not definably isomorphic, then [U_{R}(G), U_{S}(G)] = 1.

Proof – We proceed by induction on the dimension of G. We may assume that G is infinite. In particular, Z(G) is infinite (Fact 2.2) and contains a Gminimal subgroup A. Since A is G-minimal and central in G, it has no proper infinite definable subgroup. By induction hypothesis and by Proposition 3.14, the commutator [U_{R}(G), U_{S}(G)] is contained in A. We assume toward a contradiction that there exist u ∈ U_{R}(G) and v ∈ U_{S}(G) such that [u, v] is not trivial. We consider the maps f : U_{S}(G) → A and g : U_{R}(G) → A defined by f(x) = [u, x] and g(x) = [x, v]. Since [u, v] ≠ 1, they are two non-zero definable homomorphisms. Consequently, by the minimality of A and since U_{R}(G) and U_{S}(G) are definably connected, the maps f and g are surjective. Now A is both a U_{R}-group and a U_{S}- group by Proposition 3.14, and since A has no proper infinite definable subgroup, it is definably isomorphic to R_{+} and S_{+}, contradicting Lemma 2.14. Thus we obtain [U_{R}(G), U_{S}(G)] = 1. □

Lemma 3.21.. – Let R be a definable real closed field. If G is a U_{R}-group, then G′ is a homogeneous U_{R}-group.

Proof – First we show that G/Z(G) is a homogeneous U_{R}-group. Let H/Z(G) be a definable subgroup of G/Z(G). We show that H/Z(G) is a U_{R}-group. We may assume that H/Z(G) is non-trivial. Let M/Z(G) be a maximal proper definably connected definable subgroup of H/Z(G). Then H and M are definably connected (Corollary 2.7). By Proposition 3.7, the group T (H) ≤ T (G) is contained in Z(G), and H/M is not a pseudo-torus (Proposition 3.8). Then there is a normal definable subgroup N/M of H/M such that H/N is definably isomorphic to S_{+} for a definable real closed field S. By Lemma 3.13 and the maximality of M, we obtain M = N and H = U_{S}(H)M. In particular, U_{S}(G) ≥ U_{S}(H) is not central in G, and Lemma 3.20 says that the fields R and S are definably isomorphic. Thus H/M is definably isomorphic to R_{+} and G/Z(G) is a homogeneous U_{R}-group.

We show by induction on the dimension of G that G′ is a homogeneous U_{R}-group. We may assume that G is not abelian, and we consider g ∈ Z_{2}(G)\Z(G). Then the map f : G → Z(G) defined by f(x) = [g, x] is a definable group homomorphism, and Ker f contains Z(G). Hence, by the previous paragraph, its image Im f ≃ (G/Z(G))/(Ker f/Z(G)) is a non-trivial homogeneous U_{R}-subgroup of G′. Now G′/Im f is a homogeneous U_{R}-group by induction hypothesis, and the result follows from Lemma 3.17. □

Proposition 3.22.. – For any definable real closed field R, every U_{R}-group is homogeneous. In particular, such a group is torsion-free.

Proof – Let G be a U_{R}-group. We may assume that G is infinite. By Lemmas 3.17 and 3.21, we may assume that G is abelian. By Lemma 3.3, Proposition 3.8 and Fact 2.24, there is a definable o-minimal expansion of a real closed field 𝒮 = (S, <, ⋯) such that G is definably isomorphic to an 𝒮 -definable group, and there are 𝒮 -definable subgroups 1 < Z_{1} < … < Z_{m} = G where for each l ∈ {1, …, m}, the group Z_{l}/Z_{l−1} is the additive group of 𝒮.

Thus G is a U_{S}-group (Proposition 3.14). Since G is a U_{R}-group too, the fields R and S are definably isomorphic by Lemma 2.14. Let H be a definable subgroup of G. For each i ∈ {1, …, m}, the group (H ∩ Z_{i})/(H ∩ Z_{i−1}) is either trivial or definably isomorphic to Z_{i}/Z_{i−1} ≃ S_{+} ≃ R_{+}, so H is a U_{R}-group by Proposition 3.14. This proves that G is a homogeneous U_{R}-group. □

3.4. Decomposition of nilpotent groups. In this section, se state our main result on nilpotent groups (Theorem 3.29). From U_{R}-groups, we introduced Ugroups as an analogue of unipotent subgroups of algebraic groups.

Definition 3.23.. – A U-group is a nilpotent definable group G generated by U_{R_{1}}(G), …, U_{R_{k}}(G) for definable real closed fields R_{1}, …, R_{k}.

Remark 3.24.. –

• A U-group is generated by definably connected definable subgroup, so any U-group is definably connected.

• Since, for any definable real closed field R, every definable quotient group of a U_{R}-group is a U_{R}-group, every definable quotient of a U-group is a U-group.

Lemma 3.25.. – Every definable group G has a unique maximal normal U-subgroup U(G).

Proof – Let U be a maximal normal U-subgroup of G. If V is another normal U-subgroup of G, then UV is a normal nilpotent definably connected definable subgroup of G. Since U and V are U-groups, UV is a U-group too. □

Lemma 3.26.. – In a nilpotent definable group G, the subgroup U(G) contains all the U-subgroups of G.

Proof – For each definable real closed field R, the subgroup U_{R}(G) is definable, definably connected and normal in G, so U(G) contains U_{R}(G) for each definable real closed field R, and the result follows. □

Proposition 3.27.. – For every U-group G, there are finitely many definable real closed fields R_{1}, …, R_{k} such that

G = U_{R_{1}}(G) × ⋯ × U_{R_{k}}(G)

In particular, G is torsion-free. Moreover, for each definable real closed R, if R is not definably isomorphic to R_{i} for i ∈ {1, …, k}, then U_{R}(G) is trivial.

Proof – We proceed by induction on the smallest integer k such that G is generated by U_{R_{1}}(G), …, U_{R_{k}}(G) for definable real closed fields R_{1}, …, R_{k}. In particular, the fields R_{1}, …, R_{k} are not definably isomorphic. We consider H = U_{R_{1}}(G) ⋯ U_{R_{k−1}}(G). By induction hypothesis, U_{R}(G) is trivial for each real closed field R not definably isomorphic to R_{i} for i ∈ {1, …, k − 1}, and

H = U_{R_{1}}(G) × ⋯ × U_{R_{k−1}}(G)

In particular, U_{R_{k}}(H) is trivial, so we have H ∩ U_{R_{k}}(G) = 1 and G is the direct product of U_{R_{1}}(G), …, U_{R_{k}}(G).

Let R be a definable real closed field. We show that if R is not definably isomorphic to R_{i} for i ∈ {1, …, k}, then U_{R}(G) is trivial. By the previous paragraph and Proposition 3.22, the group U_{R}(G)H/H ≤ G/H ≃ U_{R_{k}}(G) is a U_{R_{k}}-group. But by Proposition 3.14, it is a U_{R}-group, hence it is trivial by Lemma 2.14, and U_{R}(G) = U_{R}(H) is trivial, as desired. □

Corollary 3.28.. – For any U-group G, we have T (G) = 1.

Proof – By Proposition 3.27, there are finitely many definable real closed fields R_{1}, …, R_{k} such that G is the direct product of U_{R_{1}}(G), …, U_{R_{k}}(G). We proceed by induction of k. By induction hypothesis, the group T (G/U_{R_{1}}(G)) ≃ T (U_{R_{2}}(G)× … × U_{R_{k}}(G)) is trivial, and Proposition 3.8 gives T (G) ≤ U_{R_{1}}(G). Then T (G) is a U_{R_{1}}-group (Proposition 3.22), so T (G) is trivial. □

Theorem 3.29.. – Any nilpotent definably connected definable group G is the central product of T (G) by U(G). More precisely, the following decomposition holds

G = T (G) ∗ (U_{R_{1}}(G) × ⋯ × U_{R_{k}}(G))

for definable real closed fields R_{1}, …, R_{k} such that U_{R}(G) = 1 for each definable real closed field not definably isomorphic to R_{i} for i = 1, …, k. Moreover, U_{R_{i}}(G) is a homogeneous U_{R_{i}}-group for each i = 1, …, k.

Proof – It follows from Proposition 3.7 that T (G) is central in G, so the group T (G)U(G) is the central product of T (G) by U(G). We assume toward a contradiction that G ≠ T (G)U(G). Let M be a maximal definably connected definable subgroup of G containing T (G)U(G). Since M contains U(G), it contains U_{R}(G) for each definable real closed field R, and Proposition 3.14 shows that no definable quotient of G/M is definably isomorphic to R_{+} for a definable real closed field R. Thus G/M is a pseudo-torus and Proposition 3.8 gives G = T (G)M, contradicting that M contains T (G). Hence we have G = T (G)U(G), and the decomposition of G follows from Propositions 3.22 and 3.27. □

Corollary 3.30.. – A nilpotent definably connected definable group G is a U-group is and only if T (G) is trivial.

Proof – This follows from Corollary 3.28 and Theorem 3.29. □

Corollary 3.31.. – Every definable subgroup H of a U-group G is a U-group.

Proof – The group G is torsion-free by Proposition 3.27, so H is definably connected and this follows from Corollary 3.30. □

Corollary 3.32.. – The derived subgroup G′ of a definably connected definable nilpotent group G is a U-group.

Proof – This follows from Theorem 3.29, Corollary 3.6 and Lemma 3.21. □

4. Structure of definable groups

The purpose of this section is to describe the structure of any definably connected definable group G from U(G). We show that G/U(G) is a central extension of a direct product of definably simple definable groups (Theorem 4.10). The proof is based on the structure of nilpotent groups (Theorem 3.29), and on the study of group actions on a solvable group.

Lemma 4.1.. – Let G be a solvable definably connected definable group. Then G′ is contained in U(G).

Proof – By Fact 2.5, the group G′ is contained in F (G). Since G is definably connected and F (G)/F (G)^{∘} is finite, G centralizes F (G)/F (G)^{∘}. Since \overline{T} = F (G)^{∘}/U(G) is a pseudo-torus by Proposition 3.8 and Theorem 3.29, the group \overline{T} is centralized by G too (Corollary 3.6). Consequently, G/U(G) is a nilpotent definably connected definable group, and by Corollary 3.32, its derived subgroup is a normal U-subgroup of F (G)/U(G).

Let R be a definable real closed field. By Proposition 3.14, we have

U_{R}(F (G)/U(G)) = U_{R}(F (G))U(G)/U(G),

and since U(G) = U(F (G)) contains U_{R}(F (G)), the groups U_{R}(F (G)/U(G)) and U(F (G)/U(G)) are trivial. But G′U(G)/U(G) is contained in U(F (G)/U(G)) by the previous paragraph, hence G′ is contained in U(G), as desired. □

Lemma 4.4 generalizes Fact 2.5. Thanks to Lemma 4.3, its proof is slightly simpler than the one of [14, Theorem 6.10]. Moreover, it uses the following result.

Fact 4.2.. – [25, Corollary 2.15 (i)] Any infinite definable group has an infinite definable abelian subgroup.

Lemma 4.3.. – Let G be a definably connected definable group. If H is a normal definable subgroup such that G/H is solvable, then G = R(G)H.

Proof – By Fact 2.4, the subgroup R generated by all normal solvable subgroups of G is definable and solvable. Then we have R(G) = R^{∘}, and G/R satisfies the hypotheses of Fact 2.8. Thus G/R is the direct product of definable subgroups H_{1}/R, …, H_{k}/R such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism between H_{i}/R and a semialgebraic subgroup of GL_{n_{i}}(R_{i}). Moreover, H_{i}^{∘}R/R is definably simple. Since G/R is definably connected, H_{i}/R is definably connected for each i. In particular, H_{i}/R = H_{i}^{∘}R/R is definably simple for each i, and G/R has no proper definable subgroup \overline{N} such that (G/R)/\overline{N} is solvable. Thus we obtain G = RH, and since G is definably connected and R(G) = R^{∘}, this implies G = R(G)H. □

Lemma 4.4.. – Let G and H be two definably connected definable groups. We assume that H is solvable. If G acts definably by conjugation on H, then [G, H] is contained in U(H).

Proof – We consider a minimal counter-example G acting on H. By minimality of G and Fact 4.2, the group \overline{G} = G/C_{G}(H/U(H)) is abelian. By Lemma 4.3, we have G = R(G)C_{G}(H/U(H)), so G = R(G) is solvable by minimality of G. We consider the semi-direct product H ⋊ G where G acts by conjugation on H. It is a solvable definably connected definable group. Then [G, H] ≤ (H ⋊ G)′ is contained in U(H ⋊ G) ∩ H by Lemma 4.1. Since U(H ⋊ G) ∩ H is a normal U-subgroup of H by Corollary 3.31, we obtain [G, H] ≤ U(H), contradicting G ≠ C_{G}(H/U(H)). Thus [G, H] is contained in U(H). □

Lemma 4.5.. – Let H be a (non-necessarily definable) subgroup of a nilpotent definable group G. Then H has a unique maximal definably connected definable subgroup.

Proof – We proceed by induction on the dimension of G. We may assume that G is definably connected. Let M be a maximal definably connected definable subgroup of H. We show that M is normal in H. We may assume that M is not normal in G. By induction hypothesis, M is the unique maximal definably connected definable subgroup of N_{H}(M) ≤ N_{G}(M) < G, therefore M is normal in N_{H}(N_{H}(M)) and we obtain N_{H}(N_{H}(M)) = N_{H}(M). But H is nilpotent, hence N_{H}(M) = H and M is normal in H.

Now, if N is any definably connected definable subgroup of H, then NM is a definably connected definable subgroup of H too, and it is contained in M by the maximality of M. This proves the uniqueness of M. □

Corollary 4.6.. – In any nilpotent definable group G, every family of definably connected definable subgroups of G generate a definably connected definable subgroup.

The following result and Corollary 4.9 are in the spirit of [1].

Proposition 4.7.. – Let G and H be two definably connected definable groups. We assume that H is solvable. If G acts definably by conjugation on H, then [G, H] is a U-subgroup of H. In particular, [G, H] is definable and definably connected.

Proof – By Lemma 4.4, the group [G, H] is contained in U(H), so we have just to prove that [G, H] is definable and definably connected. We proceed by induction on the dimension of H. Since [G, H] is contained in the nilpotent definable group U(H), it has a unique maximal definably connected definable subgroup M (Lemma 4.5). If M is nontrivial, then [G, H]/M is definable and definably connected by induction hypothesis, so [G, H] is definable and definably connected. Thus we may assume that [G, H] contains no non-trivial definably connected definable subgroup.

We show that [G, H] is central in U(H). We may assume that U(H) is nontrivial. By induction hypothesis, [G, H]Z(U(H))/Z(U(H)) is definable and definably connected. Since U(H) is a U-group, it is torsion-free (Proposition 3.27), so Z(U(H)) is definably connected and [G, H]Z(U(H)) is a definably connected definable subgroup.

• If U(H) = [G, H]Z(U(H)), then [G, H] contains U(H)′. By Corollary 3.32, the subgroup U(H)′ ≤ [G, H] is definable and definably connected, so it is trivial and U(H) is abelian.

• If [G, H]Z(U(H)) < U(H), then [U(H), [G, H]Z(U(H))] is a definably connected definable subgroup by induction hypothesis, and since it is contained in [G, H], it is trivial. Thus U(H) centralizes [G, H].

Now, for each g ∈ G, the map ad_{g} : U(H) → Z(U(H)) defined by ad_{g}(x) = [g, x] is a definable group homomorphism. Since its image is a definably connected definable subgroup of [G, H], it is trivial, so G centralizes U(H).

Thus, for each h ∈ H, the map ad_{h} : G → Z(U(H)) defined by ad_{h}(x) = [x, h] is a definable group homomorphism. Since its image is a definably connected definable subgroup of [G, H], it is trivial, we obtain [G, H] = 1 and [G, H] is a definably connected definable subgroup of H. □

Corollary 4.8.. – Let G be a definably connected definable group acting definably by conjugation on a nilpotent definable group H. Then [G, H] = [G, H^{∘}] is a Usubgroup of H.

Proof – By Fact 2.6 (1), the group H has a finite characteristic subgroup F such that H = H^{∘} ∗F. Since G is definably connected, it centralizes F and we have [G, H] = [G, H^{∘}]. Now the result follows from Proposition 4.7. □

The following result is not useful for Theorem 4.10, it will be used in the proof of Theorem 5.17.

Corollary 4.9.. – Let G be a definably connected definable group acting definably by conjugation on a solvable definable group H. Then [G, H] is a definably connected definable subgroup of H.

Proof – We proceed by induction on the dimension of the group H ⋊ G where G acts by conjugation on H. We may assume that G acts faithfully on H. If [G, H] contains a non-trivial (H ⋊ G)-normal definably connected definable subgroup A, then we may applied our induction hypothesis to H/A⋊G where G acts by conjugation on H/A, and we obtain that [G, H] is a definably connected definable subgroup of H. Thus we may assume that [G, H] contains no non-trivial (H ⋊ G)-normal definably connected definable subgroup.

The group [G, H^{∘}] is definable and definably connected by Proposition 4.7, so its H-conjugates too. Since [G, H^{∘}] is normal in H^{∘}, its H-conjugates too, so the subgroup L generated by the H-conjugates of [G, H^{∘}] is definable and definably connected. But L is a subgroup of [G, H] normal in H ⋊ G. Hence it is trivial by the previous paragraph, and G centralizes H^{∘}.

Since H/H^{∘} is finite and G is definably connected, [G, H] is contained in H^{∘}. Then for each h ∈ H, we may consider the map u_{h} : G → H^{∘} defined by u_{h}(x) = [x, h]. Since G centralizes H^{∘}, the map u_{h} is a group homomorphism, and its image is a definably connected definable subgroup of H^{∘}. Moreover, for each a ∈ H^{∘} and each x ∈ G, since G centralizes H^{∘} we have

u_{h}(x)^{a} = [x, h^{a}] = [x, a^{−1}a^{h^{−1}}h] = [x, h]

So the image of u_{h} is central in H^{∘}, and the subgroup generated by Im u_{h} for h ∈ H is a definably connected definable subgroup of Z(H^{∘}). But this subgroup is equal to [G, H], and it is normalized by G and H. Hence it is trivial by the first paragraph, and G centralizes H. Thus [G, H] = 1 is definable and definably connected. □

Theorem 4.10.. – Let G be a definably connected definable group. Then G/U(G) is a central extension of a direct product of definably simple definable groups.

More precisely, G has a normal solvable definable subgroup R such that the following three conditions hold:

• R contains all the normal solvable subgroups of G;

• [G, R] is a U-group and [G, R] = [G, R^{∘}];

• (Fact 2.8) G/R is the direct product of definably simple definable subgroups H_{1}, …, H_{k} such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism between H_{i} and a semialgebraic subgroup of GL(n_{i}, R_{i}).

Proof – By Fact 2.4, the subgroup R generated by all normal solvable subgroups of G is definable and solvable. Then we have R(G) = R^{∘}, and G/R satisfies the hypotheses of Fact 2.8. Thus, the first and the third assertions are satisfied.

Now, since [G, R^{∘}] is a U-group (Proposition 4.7), we have just to prove that [G, R] = [G, R^{∘}]. Since G is definably connected and since R/R^{∘} is finite, G centralizes R/R^{∘}. In particular, R/R^{∘} is abelian. Moreover, since [G, R^{∘}] contains [R, R^{∘}], the group R centralizes R^{∘}/[G, R^{∘}], so the group R/[G, R^{∘}] is nilpotent. Then Corollary 4.8 shows that the commutator [G/[G, R^{∘}], R/[G, R^{∘}]] is trivial and we obtain [G, R] = [G, R^{∘}]. □

5. Linearity of definable groups

We prove the main theorem in this section (Theorem 5.17). Its proof is based on the previous sections, on the study of definably linear groups (Definition 5.12 and Fact 5.4) and on the analysis of groups definable in an o-minimal expansion of a real closed field. In particular, the following two results are crucial for the proof of Theorem 5.17.

Fact 5.1.. – [20, Proof of Corollary 3.1] Let ℛ = (R, <, ⋯) be an o-minimal expansion of a real closed field. If G is definable in ℛ, then G/Z(G) can be definably embedded into GL_{n}(R).

Proposition 5.2.. – Let ℛ_{0} = (R, <, +, ·) be a definable real closed field, and let ℛ be a definable expansion of ℛ_{0} such that, for each integer n, all the definable relations of R^{n} are ℛ-definable. Let H be a normal definable subgroup of a definable group G. If H and G/H are definably isomorphic to an ℛ-definable group, then G is definably isomorphic to an ℛ-definable group.

Proof – First we assume that G^{∘} is definably isomorphic to an ℛ-definable group. We applied the method of Borovik and Cherlin [16, Proposition 4.3]. Let W be the wreath product of G^{∘} by G/G^{∘}. It is definably isomorphic to an ℛ-definable group, and we have just to find a definable group monomorphism from G to W. We consider a left transversal T = {g_{1}, …, g_{r}} to G^{∘} in G. For each x ∈ G and each i ∈ {1, …, r}, we denote by n_{i}(x) the unique element of G^{∘} such that n_{i}(x)g_{i}x ∈ T, and we define a map μ : G → W by μ(x) = ((n_{1}(x), …, n_{r}(x)), xG^{∘}). The map μ is definable, and it is a group homomorphism (see the proof of [9, Theorem 18.9 p.68]). Moreover, if x belongs to Ker μ, the last coordinate gives x ∈ G^{∘}, and since g_{1}x ∈ {g_{1}, …, g_{r}}, we obtain g_{1}x = g_{1} and x = 1. Thus μ is a definable group monomorphism from G to W, as desired. Hence we may assume that G is definably connected.

Now we proceed by induction on the dimension of H. By the structure of H^{∘} described in Theorem 4.10, we may assume that either H is finite, or H has no non-trivial normal abelian subgroup, or H is abelian, or H is a U-group. Suppose that H has no non-trivial normal abelian subgroup. By Theorem 4.10, the group G has a normal solvable definable subgroup R such that the following two conditions hold:

• R contains all the normal solvable subgroups of G;

• G/R is the direct product of definably simple definable subgroups. Consequently, since HR/R is a normal definable subgroup of G/R, the group HR/R is a direct product of some subgroups H_{1}, …, H_{k}, and G/R has a normal definable subgroup S/R such that G/R = HR/R × S/R. Thus we have G = HS and H ∩ S ≤ R. But R is solvable and H has no non-trivial normal abelian subgroup, so R ∩ H is trivial, and since H and S are normal in G, we obtain G = H × S. Hence G ≃ H × G/H is definably isomorphic to an ℛ-definable group. Thus we may assume that either H is finite, or H is abelian, or H is a U-group.

However, if H is finite, then since G is definably connected, G centralizes H, and H is abelian. Moreover, if H is a non-abelian U-group, then H′ is infinite and definable (Corollary 3.32), and the induction hypothesis applied with H/H′ and H′ shows that G is definably isomorphic to an ℛ-definable group. Hence we may assume that H is abelian.

For each g ∈ G, we denote by \overline{g} = gH the left coset of g modulo H. By Fact 2.1, there is a definable function t : G → G such that for all x, y ∈ G, we have x ∈ xH and if xH = yH then t(x) = t(y). We define a map Φ : \overline{G} × \overline{G} → H by Φ(xH, yH) = t(xy)^{−1}t(x)t(y). In particular, the map Φ is definable, so its graph is a definable subset of \overline{G} × \overline{G} × H ⊆ R^{n} for an integer n, and Φ is ℛ-definable by our hypothesis over ℛ. We consider the set L = G/H × H and the group ℒ = (L, ⊗) where for every (\overline{g}, h) ∈ L and (\overline{g′}, h′) ∈ L, the product (\overline{g}, h) ⊗ (\overline{g′}, h′) is defined by

(\overline{g}, h) ⊗ (\overline{g′}, h′) = (\overline{g}\overline{g′}, h^{g′}h′Φ(\overline{g}, \overline{g′}))

We note that, since the groups G/H and H are definably isomorphic to ℛ-definable groups, and since Φ is ℛ-definable too, the group ℒ = (L, ⊗) is definably isomorphic to an ℛ-definable group. Moreover the map f : G → L defined by f(g) = (\overline{g}, t(g)^{−1}g) is a definable group isomorphism, so G is definably isomorphic to an ℛ-definable group. □

Corollary 5.3.. – Let ℛ_{0} = (R, <, +, ·) be a definable real closed field, and let ℛ be a definable expansion of ℛ_{0} such that, for each integer n, all the definable relations of R^{n} are ℛ-definable. Then every U_{R}-group is definably isomorphic to an ℛ-definable group.

Proof – This follows from Propositions 3.22 and 5.2. □

The definable subgroups of GL_{n}(R) are studied in [22] whose main result is Fact 5.4. We provide below some useful complements. In particular, we show that a definable quotient of a definably connected subgroup of GL_{n}(R) is definably isomorphic to a subgroup of GL_{n}(R) (Proposition 5.11).

Fact 5.4.. – [22, Theorem 4.1] Let ℛ be an o-minimal expansion of a real closed field (R, <, ⋯), and let G be a ℛ-definably connected ℛ-definable subgroup of GL_{n}(R) for an integer n. Then there are semialgebraic subgroups G_{1} and G_{2} of GL_{n}(R) such that G_{2} ≤ G ≤ G_{1}, G_{2} is a normal subgroup of G_{1} and G_{1}/G_{2} is abelian. Moreover, there are abelian, ℛ-definable, ℛ-definably connected subgroups A_{1}, …, A_{k} of G such that G = G_{2} · A_{1} ⋯ A_{k}.

Fact 5.5.. – [22, Lemma 3.4 (ii)] Let R be a definable real closed field, and let G be a definable subgroup of GL_{n}(R) for an integer n. If G is a definable subgroup of a semialgebraic group of the form (R_{+})^{k}, then G is semialgebraic.

Corollary 5.6.. – Let R be a definable real closed field, and let G be a definable subgroup of GL_{n}(R) for an integer n. If G is a definable subgroup of a semialgebraic unipotent group U, then G is semialgebraic.

Proof – We proceed by induction on the dimension of G. Since U is unipotent, it is torsion-free and its definable subgroups are definably connected. Let M be a maximal proper definable subgroup of G. By induction hypothesis, M is semialgebraic. Then G/M is a definable subgroup of the semialgebraic unipotent group N_{U}(M)/M. Thus, if M is non-trivial, G/M is semialgebraic by induction hypothesis, so G is semialgebraic. Hence we may assume that M is trivial. Now G is abelian (Fact 4.2), and it is a definable subgroup of Z(C_{U}(G)). Since Z(C_{U}(G)) is an abelian semialgebraic unipotent group, it is of the form (R_{+})^{k}, and G is semialgebraic by Fact 5.5. □

We recall that a semisimple group is defined to be a definably connected definable group with no infinite abelian normal subgroup (Definition 6.1).

Fact 5.7.. – [22, Theorem 4.5] Let R be a definable real closed field, and let G be a definably connected definable subgroup of GL_{n}(R) for an integer n. Then G = NH for a normal solvable definable subgroup N and a semialgebraic semisimple subgroup H such that N ∩ H is finite.

Lemma 5.8.. – Any semisimple group S is perfect and satisfies R(S) = 1.

Proof – Since S is a semisimple group, Z(U(S)) is finite, and U(S) is finite too by Fact 2.2. But U(S) is a U-group, so it is definably connected, and consequently it is trivial. Hence R(S) is abelian (Proposition 4.7), and since S is semisimple, R(S) is finite. Thus, since R(S) is definably connected, R(S) is trivial.

Let R be the subgroup of S generated by all normal solvable subgroups of G. It is definable and solvable (Fact 2.4), so R^{∘} = R(S) is trivial and R is finite. Now S/R has no non-trivial normal abelian subgroup, and Fact 2.8 with Proposition 2.9 shows that S/R is perfect. Thus we have S = S′R and S′ has finite index in S, so Proposition 2.9 gives S = S′. □

Lemma 5.9.. – Let R be a definable real closed field, and let G be a definably connected definable subgroup of GL_{n}(R) for an integer n. Then G′ = U ⋊ S for a semialgebraic unipotent group U and a semialgebraic semisimple group S. In particular, G′ is semialgebraic.

Proof – By Fact 5.7, we have G = NH for a normal solvable definable subgroup N and a semialgebraic semisimple subgroup H such that N ∩H is finite. Since G is definably connected, we may assume that N is definably connected. Moreover, we have R(H) = 1 and H is perfect (Lemma 5.8). Thus we have G′ = N′[H, N]H′ = N′[H, N]H, and U = N′[H, N] is a U-group by Proposition 4.7. In particular, U is a definable subgroup of N, and it is torsion-free (Proposition 3.27), so U ∩H ∩N ∩H is trivial.

Let \overline{N} be the smallest semialgebraic subgroup of GL_{n}(R) containing N. Then \overline{N} is semialgebraically connected by Proposition 2.9. Moreover, \overline{N}′ is contained in N by Fact 5.4, so \overline{N} is solvable. This implies that \overline{N}′ is a semialgebraic unipotent group. Since H is definably connected, it is semialgebraically connected, and then [H, \overline{N}] is a semialgebraic unipotent group too. Now \overline{N}′[H, \overline{N}]H is a semialgebraic unipotent group. Since U = N′[H, N] is a definable subgroup of \overline{N}′[H, \overline{N}], Corollary 5.6 shows that U is semialgebraic, and we have the decomposition G′ = U ⋊ S with S = H. □

Corollary 5.10.. – Let R be a definable real closed field, and let G be a definably connected definable subgroup of GL_{n}(R) for an integer n. Then any normal definable subgroup H of G′ is semialgebraic.

Proof – We may assume that H is definably connected. By Fact 5.7, we have H = NT for a normal solvable definable subgroup N and a semialgebraic semisimple subgroup T such that N ∩ T is finite. Since H is definably connected, we have H = N^{∘}T. Moreover, by Lemma 5.9, there are a semialgebraic unipotent group U and a semialgebraic semisimple group S such that G′ = U ⋊ S. In particular, since R(S) = 1 (Lemma 5.8), we have R(G′) = U. Thus N^{∘} is a definable subgroup of R(H) ≤ R(G′) = U, and Corollary 5.6 implies that N^{∘} is semialgebraic. So H = N^{∘}T is semialgebraic. □

Proposition 5.11.. – Let R be a definable real closed field, and let G be a definably connected definable subgroup of GL_{n}(R) for an integer n. If H is a normal definable subgroup of G, then G/H is definably isomorphic to a definable subgroup of GL_{m}(R) for an integer m.

Proof – By Corollary 5.10, the group H ∩ G′ is semialgebraic, so N_{GL_{n}(R)}(H ∩ G′)/(H ∩ G′) definably embeds into the GL_{l}(R) for an integer l, and G/(H ∩ G′) ≤ N_{GL_{n}(R)}(H ∩ G′)/(H ∩ G′) is definably isomorphic to a definable subgroup of GL_{l}(R). Thus we may assume that H ∩ G′ is trivial.

By Lemma 5.9, the group G′ is semialgebraic, so N_{GL_{n}(R)}(G′)/G′ definably embeds into GL_{k}(R) for an integer k, and G/G′ ≤ N_{GL_{n}(R)}(G′)/G′ is definably isomorphic to a definable subgroup of GL_{k}(R). But H^{∘}G′/G′ has a definable complement C/G′ in G/G′ by Fact 2.13 (2), and since C ∩ H^{∘} ≤ G′ ∩ H is trivial, G/H^{∘} is definably isomorphic to H^{∘}C/H^{∘} ≃ C/(C ∘ H^{∘}) ≃ C. Hence we may assume that H is finite. Then N_{GL_{n}(R)}(H)/H definably embeds into GL_{m}(R) for an integer m, and G/H ≤ N_{GL_{n}(R)}(H)/H is definably isomorphic to a definable subgroup of GL_{m}(R), as desired. □

Definition 5.12.. – A definable group G is said to be definably linear (over finitely many definable real closed fields R_{1}, …, R_{k}), if G has a definable faithful linear representation over the ring R_{1} ⊕ ⋯ ⊕ R_{k}.

In other words, G definably embeds in H_{1} × ⋯ × H_{k}, where H_{i} is a linear semialgebraic group over R_{i} for each i = 1, …, k.

Lemma 5.13.. – Let R_{1}, …, R_{k} be finitely many definable real closed fields. Any definable group G has a smallest normal definable subgroup N such that G/N is definably linear over R_{1}, …, R_{k}.

Proof – It is sufficying to show that, if A and B are two normal definable subgroups of G such that G/A and G/B are definably linear, then G/(A ∩ B) is definably linear. Moreover, we may assume that A ∩ B is trivial. We consider definable real closed fields R_{1}, …, R_{k}, S_{1}, …, S_{l} such that G/A definably embeds in H_{1} × ⋯ × H_{k}, where H_{i} is a linear semialgebraic group over R_{i} for each i = 1, …, k, and such that G/B definably embeds in K_{1} ×⋯×K_{l}, where K_{j} is a linear semialgebraic group over S_{j} for each j = 1, …, l. Let f : G → G/A × G/B be the map defined by f(x) = (xA, xB). Since f is a definable group monomorphism, G is definably linear. □

Lemma 5.14.. – Let R_{1}, …, R_{k} be finitely many definable real closed fields, and let H_{i} be a definable subgroup of a linear semialgebraic group over R_{i} for each i = 1, …, k. If R_{1}, …, R_{k} are not definably isomorphic, then for any definably connected definable subgroup L of H_{1} × ⋯ × H_{k}, we have

L = (L ∩ H_{1}) × ⋯ × (L ∩ H_{k})

In particular, if G is a definably connected definable group, and if G is definably linear over finitely many definable real closed fields R_{1}, …, R_{k}, then G is definably isomorphic to a direct product of definable subgroups of GL_{n_{1}}(R_{1}), …, GL_{n_{k}}(R_{k}).

Proof – Since G is definably linear over R_{1}, …, R_{k}, the group G definably embeds into a direct product H_{1} × ⋯ × H_{k}, where H_{i} is a linear semialgebraic group over R_{i} for each i = 1, …, k. It is sufficying to show that, for any definably connected definable subgroup L of H_{1} × ⋯ × H_{k}, we have

L = (L ∩ H_{1}) × ⋯ × (L ∩ H_{k})

We assume toward a contradiction that L is a counter-example of minimal dimension. Therefore, for each proper definably connected definable subgroup L_{0} of L we have L_{0} = (L_{0} ∩ H_{1}) × ⋯ × (L_{0} ∩ H_{k}). This implies that, if we consider K = (L ∩ H_{1}) × ⋯ × (L ∩ H_{k}), then K^{∘} contains all the proper definably connected definable subgroups of L.

For each i ∈ {1, …, k}, we denote by p_{i} : H_{1} × ⋯ × H_{k} → H_{i} the i^{th} projection map. Since K is proper in L, there is i ∈ {1, …, k} such that p_{i}(L) is non-trivial. If p_{j}(L) is trivial for each j ≠ i, then L is contained in H_{i}, contradicting K < L. Therefore there exists j ≠ i such that p_{j}(L) is non-trivial. We consider K_{i} = Ker p_{i} and K_{j} = Ker p_{j}. They are proper subgroups of L, so K contains K_{i}^{∘} and K_{j}^{∘}, and K_{i}K_{j} is a proper normal definable subgroup of L. But L/K_{i} and L/K_{j} are definably isomorphic to p_{i}(L) ≤ H_{i} and p_{j}(L) ≤ H_{j} respectively, so there exists two integer m and n such that L/K_{i}K_{j} is definably isomorphic to a definable subgroup P_{i} of GL_{m}(R_{i}) and to a definable subgroup P_{j} of GL_{n}(R_{j}). Hence the fields R_{i} and R_{j} are definably isomorphic by Proposition 2.15, contradicting that the fields R_{1}, …, R_{k} are not definably isomorphic. □

Lemma 5.15.. – Let G be a definably connected definable group. If U(Z(G)) is trivial, then G/Z(G) is centerless.

Proof – We consider Z/Z(G) = Z(G/Z(G)). The subgroup Z is definable, nilpotent and normal in G. By Corollary 4.8, the group [G, Z] is a U-group, and since it is contained in Z(G), the subgroup Z is central in G. Thus G/Z(G) is centerless. □

Lemma 5.16.. – Let G be a definable group and let R be a definable real closed field. Then G has a smallest normal definable subgroup K such that G/K is a U_{R}-group.

Proof – We have just to prove that if A and B are two normal definable subgroups such that G/A and G/B are U_{R}-groups, then G/(A ∩ B) is a U_{R}-group. Since G/A and G/B are nilpotent, G/(A ∩ B) is nilpotent too. But AB/B is a definable subgroup of G/B, so it is a homogeneous U_{R}-group by Proposition 3.22, and by Proposition 3.22 again, G/A is a homogeneous U_{R}-group. Hence, since A/(A ∩ B) ≃ AB/B, it follows from Lemma 3.17 that G/(A ∩ B) is a homogeneous U_{R}-group. □

Theorem 5.17.. – Let G be a definably connected definable group. Then G/Z(G) is the direct product of definable groups \overline{H_{1}}, …, \overline{H_{k}} such that for every i ∈ \{1, …, k\} there is a definable real closed field R_{i}, an integer n_{i} and a definable isomorphism from \overline{H_{i}} to a definable subgroup of GL_{n_{i}}(R_{i}).

Proof – By Lemma 5.14, we have just to prove that G/Z(G) is definably linear. We proceed by induction on the dimension of G. By Fact 2.12, we may assume G/Z(G) is not centerless. In particular, U(Z(G)) is non-trivial by Lemma 5.15. Let R be a definable real closed field such that U_{R}(Z(G)) is non-trivial, and A be a G-minimal subgroup of U_{R}(Z(G)) (Definition 3.18). In particular, A is torsionfree (Proposition 3.22). Since A is G-minimal and central in G, it has no proper non-trivial subgroup, and since it is a U_{R}-group, it is definably isomorphic to R_{+}.

We show that we may assume that A is the unique G-minimal subgroup of G. Indeed, if G has another G-minimal subgroup B ≠ A, we consider Z_{A}/A = Z(G/A) and Z_{B}/B = Z(G/B). By induction hypothesis, the groups G/Z_{A} and G/Z_{B} are definably linear, so G/(Z_{A} ∩ Z_{B}) is definably linear by Lemma 5.13. Since A and B are G-minimal, the group A ∩ B is finite, and since G is definably connected, A ∩ B is central in G. Thus, for each z ∈ Z_{A} ∩ Z_{B}, the map u_{z} : G → A ∩ B is a definable group homomorphism, and since G is definably connected, its image is a definably connected subgroup of the finite subgroup A ∩ B. Therefore z is centralizes G and Z_{A} ∩ Z_{B} = Z(G). Now G/Z(G) is definably linear, and we may assume that A is the unique G-minimal subgroup of G. In particular, U(G) is a U_{R}-group (Proposition 3.27).

Let Z/A = Z(G/A). We show that Z = U_{R}(Z)Z(G) and that Z/Z(G) is a U_{R}- group. By induction hypothesis, the group G/Z is definably linear. For each g ∈ G, the map u_{g} : Z → A defined by u_{g}(x) = [g, x] is a group homomorphism, and since A ≃ R_{+}, we have either Ker u_{g} = Z or Z/Ker u_{g} ≃ R_{+}. It follows from Lemma 5.16 that Z/Z(G) is a U_{R}-group. Then, by Proposition 3.14, we have Z = U_{R}(Z)Z(G).

Since G/Z is definably linear, Lemma 5.14 says that G is definably isomorphic to a direct product K_{1}/Z × ⋯ × K_{k}/Z, where K_{i}/Z is a definably linear group over a definable real closed field R_{i} for each i = 1, …, k. Since G is definably connected, K_{i}/Z is definably connected for each i = 1, …, k. Moreover, we may assume that R = R_{1}, and that the fields R_{1}, …, R_{k} are not definably isomorphic.

We note that we does not say that K_{1}/Z is non-trivial. We show that [K_{1}, K_{j}] is contained in Z(G) for each j ≠ 1. For each g ∈ K_{1} and each j ≠ 1, the map \overline{ad_{j}}_{g} : K_{j} → Z/Z(G) defined by \overline{ad_{j}}_{g}(x) = [g, x] is a group homomorphism, and K_{j}/Ker \overline{ad_{j}}_{g} is definably isomorphic to a subgroup of Z/Z(G), so it is a U_{R}-group (Proposition 3.22). Since Ker \overline{ad_{j}}_{g} contains Z, either K_{j} = Ker \overline{ad_{j}}_{g}, or the group K_{j}/Z has a normal definable subgroup N/Z such that (K_{j}/Z)/(N/Z) is definably isomorphic to R_{+}. In the second case, since (K_{j}/Z)/(N/Z) is definably isomorphic to a definable linear group over R_{j} by Proposition 5.11, the fields R_{j} and R = R_{1} are definably isomorphic by Proposition 2.15, contradicting j ≠ 1. Thus we have K_{j} = Ker \overline{ad_{j}}_{g}, and [K_{1}, K_{j}] is contained in Z(G) for each j ≠ 1.

Let j ≠ 1 and let H_{j}/Z(G) be a definable subgroup of K_{j}/Z(G) such that K_{j} = ZH_{j}, and minimal for this condition. We prove that K_{1} centralizes H_{j}. Since Z/Z(G) is a U_{R}-group, it is definably connected, and since K_{j}/Z is definably connected too, the group K_{j}/Z(G) is definably connected, so such a subgroup H_{j}/Z(G) is definably connected. By the previous paragraph, for each g ∈ K_{1} and each j ≠ 1, we may define a group homomorphism ad_{j}_{g} : H_{j} → Z(G) by ad_{j}_{g}(x) = [g, x]. Therefore H_{j}/Ker ad_{j}_{g} is definably isomorphic to the subgroup Im ad_{j}_{g} of Z(G). In particular, H_{j}/Ker ad_{j}_{g} is abelian. Since Ker ad_{j}_{g} contains Z(G) and since H_{j}/Z(G) is definably connected, Lemma 4.3 gives H_{j} = R_{j}Ker ad_{j}_{g} where R_{j}/Z(G) = R(H_{j}/Z(G)). Now R_{j} is a normal solvable subgroup of G, and [G, R_{j}] is a U-group (Corollary 3.31 and Theorem 4.10). Thus, since U(G) is a U_{R}- group by the second paragraph, [G, R_{j}] and Im ad_{j}_{g} are U_{R}-subgroups (Proposition 3.22). So, if Im ad_{j}_{g} ≃ H_{j}/Ker ad_{j}_{g} is non-trivial, then H_{j}/Ker ad_{j}_{g} has a proper normal definable subgroup N/Ker ad_{j}_{g} such that H_{j}/N is definably isomorphic to R_{+}. By minimality of H_{j}, we have K_{j} ≠ ZN , so we obtain (Z ∩ H_{j})N < H_{j}, and since H_{j}/N ≃ R_{+} has no non-trivial proper definable subgroup (Fact 2.10), we have Z ∩ H_{j} ≤ N. Thus we obtain

(K_{j}/Z)/(NZ/Z) ≃ K_{j}/NZ = H_{j}Z/NZ ≃ H_{j}/(H_{j} ∩ NZ) = H_{j}/N ≃ R_{+}

Now, by Propositions 2.15 and 5.11, the fields R_{1} = R and R_{j} are definably isomorphic, contradicting j ≠ 1. Consequently Im ad_{j}_{g} is trivial and every g ∈ K_{1} centralizes H_{j}.

In particular, the previous paragraph shows that H_{2}, … H_{k} centralize Z ≤ K_{1}, and since G = K_{1}H_{2} ⋯ H_{k}, we obtain C_{Z}(K_{1}) = C_{Z}(G) = Z(G). Then, for each j = 2, … k, we have H_{j} ∩ Z ≤ C_{Z}(K_{1}) = Z(G), and by the previous paragraph again, we have C_{K_{j}}(K_{1}) = C_{ZH_{j}}(K_{1}) = H_{j}C_{Z}(K_{1}) = H_{j}Z(G) = H_{j}. Consequently H_{j} is a normal definable subgroup of G. Moreover, we note that Z ∩ (H_{2} ⋯ H_{k}) is contained in Z ∩ C_{G}(K_{1}) = Z(G). Thus, since G/Z is the direct product of K_{1}/Z, … K_{k}/Z, and since K_{i} = ZH_{i} for each i = 2, …, k, we obtain

G/Z(G) = K_{1}/Z(G) × H_{2}/Z(G) × ⋯ × H_{k}/Z(G)

But for each i = 2, …, k, the group H_{i}/Z(G) = H_{i}/(H_{i} ∩ Z) ≃ H_{i}Z/Z = K_{i}/Z is definably linear over R_{i}. Hence we have just to prove that K_{1}/Z(G) is definably linear over R.

Let U/U_{R}(G) = U_{R}(G/U_{R}(G)). We show that C_{Z}(U) = Z(G). For each z ∈ Z, we consider the definable group homomorphism v_{z} : G → A defined by v_{z}(x) = [x, z]. Since A is definably isomorphic to R_{+}, the group G/Ker v_{z} ≃ Im v_{z} is a U_{R}-group for each z ∈ Z, and G/C_{G}(Z) is a U_{R}-group by Lemma 5.16. Moreover, Lemma 4.3 shows that G = R(G)C_{G}(Z). In particular, R(G)/(R(G)∩C_{G}(Z)) is a nilpotent group. Let D = U_{R}(R(G) ∩ C_{G}(Z)). Since R(G)/U(G) is abelian (Proposition 4.7) and since U(G) is a U_{R}-group, the group [R(G), R(G) ∩ C_{G}(Z)] is a definable subgroup of U_{R}(G) (Corollary 4.9), and it is contained in D (Proposition 3.22). This implies that R(G)/D is a nilpotent group. Since R(G)/(R(G) ∩ C_{G}(Z)) ≃ G/C_{G}(Z) is a U_{R}-group and D is contained in R(G) ∩ C_{G}(Z), Proposition 3.14 shows that the subgroup V defined by V/D = U_{R}(R(G)/D) covers R(G)/(R(G) ∩ C_{G}(Z)) and G/C_{G}(Z). Moreover, U_{R}(G) contains D and, since U_{R}(G)/D is a U_{R}-group (Proposition 3.14), the group V/D contains U_{R}(G)/D, and V/U_{R}(G) ≃ (V/D)/(U_{R}(G)/D) is a normal U_{R}-subgroup of G/U_{R}(G). Thus V is contained in U, and we obtain G = V C_{G}(Z) = UC_{G}(Z), so C_{Z}(U) = Z(G).

Let ℛ be a definable expansion of (R, <, +, ·) such that, for each integer n, all the definable relations of R^{n} are ℛ-definable. By Proposition 5.2 and Corollary 5.3, the groups U_{R}(G), U and Z/Z(G) = U_{R}(G)Z(G)/Z(G) are definably isomorphic to ℛdefinable groups. Moreover, since K_{1}/Z is definably linear over R, Proposition 5.2 says that K_{1}/Z(G) is definably isomorphic to an ℛ-definable group. We consider the semi-direct product L = U ⋊ K_{1}/Z(G) where K_{1}/Z(G) acts by conjugation on U. Then, by Proposition 5.2 again, the group L is definably isomorphic to an ℛ-definable group. Now L/Z(L) is definably linear over L (Fact 5.1). Let N/Z(G) = Z(L) ∩ K_{1}/Z(G). Then K_{1}/N is definably linear over R. Moreover, since C_{Z}(U) = Z(G) by the previous paragraph, Z(L) ∩ Z/Z(G) is trivial, so N ∩ Z = Z(G). Since K_{1}/Z and K_{1}/N are definably linear over R, Lemma 5.13 shows that K_{1}/Z(G) is definably linear over R, and we conclude that G/Z(G) is definably linear over R. □

We may state Theorem 5.17 under the following formulation.

Corollary 5.18.. – Let G be a definably connected definable group. Then G is the central product of definable subgroups H_{1}, …, H_{k} such that for every i ∈ {1, …, k} there is a definable real closed field R_{i}, an integer n_{i} and a definable isomorphism from H_{i}Z(G)/Z(G) to a definable subgroup of GL_{n_{i}}(R_{i}).

Proof – We may assume that G is ω-saturated. We consider the groups H_{i}/Z(G) = \overline{H_{i}} in Theorem 5.17, and we assume that the fields R_{1}, …, R_{k} are not definably isomorphic. We have just to prove that the groups H_{i}/Z(G) = \overline{H_{i}} in Theorem 5.17 satisfy [H_{i}, H_{j}] = 1 for each j ≠ i. By Facts 2.6 (1) and 4.2, there are a ∈ H_{i} and b ∈ H_{j} such that [a, b] is of infinite order. We consider the maps u : H_{j} → Z(G) defined by u(x) = [a, x] and v : H_{i} → Z(G) defined by v(x) = [x, b]. They are definable group homomorphisms, and since I = Im u∩Im v contains [a, b], the group I is infinite. But Ker u (resp. Ker v) contains Z(G), so Im u (resp. Im v) is definably isomorphic to a definable quotient of H_{i}/Z(G) (resp. H_{j}/Z(G)) which is definably isomorphic to a definable subgroup of GL_{n_{i}}(R_{i}) (resp. GL_{n_{j}}(R_{j})). This implies that I is an infinite definable group which is, by Proposition 5.11, definably linear over R_{i} and definably linear over R_{j}. Hence, by Proposition 2.15, the fields R_{i} and R_{j} are definably isomorphic, contradicting our hypothesis over the fields R_{1}, …, R_{k}. □

Corollary 5.19.. – Let G be a definably connected definable group. Then the group G′Z(G)/Z(G) is a definably connected definable subgroup. More precisely, G′Z(G)/Z(G) is a direct product of definably connected definable groups \overline{H_{1}}, …, \overline{H_{k}} such that for every i ∈ {1, …, k} there is a definable real closed field R_{i} and a definable isomorphism between \overline{H_{i}} and a semialgebraic linear group over R_{i}.

Proof – This follows from Theorem 5.17 and Lemma 5.9. □

6. A Levi-like decomposition

Conversano exhibited a definably connected definable group G such that R(G) = Z(G) and whose derived subgroup is not definable [5, Example 3.1.7]. Moreover, this group G has no semisimple subgroup S such that G = R(G)S. This motivates the introduction of quasi-semisimple groups.

Definition 6.1.. – Let S be a definably connected definable subgroup of a definable group G.

• S is said to be semisimple if it has no infinite abelian normal subgroup;

• S is said to be quasi-semisimple if R(S)H < S for every proper definable subgroup H of S.

Remark 6.2.. – It follows from Lemma 5.8 that any semisimple group S is quasisemisimple.

Conversano and Pillay introduce in [6] ind-definable semisimple subgroups, and they show their existence and conjugacy in every definably connected group G definable in an o-minimal expansion ℛ of a real closed field.

We refer to [6] for the definition of an ind-definable semisimple subgroup, and we provide just their main properties.

Fact 6.3.. – [6, Theorem 1.1] Let ℛ be an o-minimal expansion of a real closed field K, and let G be an ℛ-definably connected ℛ-definable group. Then G has a maximal ind-definable semisimple subgroup S, unique up to conjugacy in G. Moreover G = R(G)S, and the centre Z(S) of S is finitely generated and contains R(G) ∩ S.

Furthermore, the following properties are satisfied:

(1) [6, Lemma 2.7] any ind-definable semisimple subgroup of G is perfect;

(2) [6, Proof of Theorem 1.1] there is a maximal semisimple subgroup T/Z(G)^{∘} of G/Z(G)^{∘} such that S = T′.

(3) [6, Proofs of Lemmas 4.1 and 4.2] if G is a definable subgroup of GL_{n}(K) for an integer n, the maximal ind-definable semisimple subgroups of G are precisely its maximal semisimple subgroups.

We will show that, if S is a subgroup of an ℛ-definably connected ℛ-definable group, then S is a maximal ind-definable semisimple subgroup if and only if it is the derived subgroup of a maximal quasi-semisimple subgroup (Corollary 6.7).

Lemma 6.4.. – Let G be a definably linear definable group. If G is definably connected, then G has a maximal semisimple subgroup S, unique up to conjugacy in G. Moreover, G = R(G)S and R(G) ∩ S is finite and contained in the centre of S.

Proof – By Lemma 5.14, there are finitely many definable real closed fields R_{1}, …, R_{k} such that G is definably isomorphic to a direct product H_{1} × ⋯ × H_{k}, where H_{i} is a definable subgroup of a linear algebraic group over R_{i} for each i = 1, …, k. By Fact 5.7, for each i, we find in H_{i} a maximal semisimple subgroup S_{i} such that H_{i} = R(H_{i})S_{i} and R(H_{i}) ∩ S_{i} is finite. Then we have R(G) = R(H_{1})×⋯×R(H_{k}), and S = S_{1} ×⋯×S_{k} is a semisimple subgroup. In particular, we obtain G = R(G)S, and R(G) ∩ S is finite. Moreover, since S is definably connected, it centralizes the finite normal subgroup R(G) ∩ S.

Let T be a maximal semisimple subgroup of G. By Lemma 5.14, we have

T = (T ∩ H_{1}) × ⋯ × (T ∩ H_{k})

In particular, for each i, the subgroup T ∩ H_{i} is a maximal semisimple subgroup of H_{i}, and by Fact 6.3, the subgroups S_{i} and T ∩ H_{i} are conjugate in H_{i}. Thus S and T are conjugate in G. □

Corollary 6.5.. – Let G be a definably linear definable group. Then G is semisimple if and only if it is quasi-semisimple.

Proof – By Remark 6.2, we may assume that G is a quasi-semisimple group, and we have just to prove that G is semisimple. By Lemma 6.4, the group G has a semisimple subgroup S such that G = R(G)S. Since G is quasi-semisimple, this implies that G = S is semisimple. □

For each subset X of a definable group G, the intersection of all definable subgroups of G containing X is a definable subgroup by descending chain condition on definable subgroups [25, Remark 2.13 (ii)]. This subgroup is denoted by d(X).

Theorem 6.6.. – Let G be a definably connected definable group. Then G has a maximal quasi-semisimple subgroup S, unique up to conjugacy in G. Moreover

• G = R(G)S;

• R(G) ∩ S is central in S.

Moreover, SZ(G)/Z(G) is a maximal semisimple subgroup of G/Z(G), S′ is a perfect group, S = d(S′), and S/Z(S) has no non-trivial normal abelian subgroup.

Proof – By Theorem 5.17, the group G/Z(G) is definably linear. By Corollary 6.5, its semisimple subgroups are precisely its quasi-semisimple subgroups. By Lemma 6.4, it has a maximal quasi-semisimple subgroup S_{0}/Z(G), unique up to conjugacy in G/Z(G). Moreover, we have

G/Z(G) = R(G/Z(G))S_{0}/Z(G)

and R(G/Z(G)) ∩ S_{0}/Z(G) is contained in the (finite) centre of S_{0}/Z(G), and by Lemma 5.8, the subgroup S_{0}/Z(G) is perfect.

We consider S = d(S′_{0}). Since S_{0}/Z(G) is perfect, we have S_{0} = S′_{0}Z(G) = S′′_{0}Z(G), so S_{0} = SZ(G) and S′_{0} = S′. In particular, we have S = d(S′). Moreover, since S_{0}/S′′_{0} = Z(G)S′′_{0}/S′′_{0} is abelian, we obtain S′ = S′_{0} = S′′_{0} = S′′ and S′ is perfect.

We show that S is a quasi-semisimple subgroup and that R(S) is contained in Z(G) ∩ S. Since SZ(G)/Z(G) = S_{0}/Z(G) is quasi-semisimple, it is definably connected, and we have S = S^{∘}(S ∩ Z(G)). Therefore S′_{0} = S′ = (S^{∘})′ is contained in S^{∘}, and S = d(S′_{0}) is contained in S^{∘} too, so S is definably connected. Since S/(Z(G) ∩ S) ≃ SZ(G)/Z(G) = S_{0}/Z(G) is semisimple, the radical R(S/(Z(G) ∩ S)) is trivial (Lemma 5.8) and R(S) is contained in Z(G) ∩ S. Thus, if H is a definable subgroup of S such that R(S)H = S, then we have (Z(G) ∩ S)H = S and H′ = S′ = S′_{0}. This implies that H contains S = d(S′_{0}) = d(H′), so H = S and S is quasi-semisimple.

We show that any quasi-semisimple subgroup of G is contained in a conjugate of S. Let T be such a subgroup. We may assume that no quasi-semisimple subgroup of G contains properly T. If H/Z(G) is a definable subgroup of T Z(G)/Z(G) such that R(T Z(G)/Z(G))H/Z(G) = T Z(G)/Z(G), then we have

T Z(G) = R(T Z(G))H = R(T)H

and T = R(T)(T ∩ H), so T ∩ H = T because T is quasi-semisimple. Therefore H contains T, we have H/Z(G) = T Z(G)/Z(G), and T Z(G)/Z(G) is quasisemisimple. Now T Z(G)/Z(G) is a semisimple subgroup of G/Z(G), and it is contained in a conjugate of S_{0}/Z(G) = SZ(G)/Z(G) by Lemma 6.4, so we may assume that T Z(G)/Z(G) is contained in SZ(G)/Z(G). In particular, we have T′ = (T Z(G))′ ≤ (SZ(G))′ = S′. But T Z(G)/Z(G) is a semisimple group, so it is perfect (Lemma 5.8), and we obtain T Z(G) = T′Z(G) and

T = T′(T ∩ Z(G)) = d(T′)(T ∩ Z(G))^{∘} = d(T′)R(T)

Hence, since T is quasi-semisimple, we have T = d(T′) and T is contained in d(S′) ≤ S, as desired.

We show that S/Z(S) has no non-trivial normal abelian subgroup. If A/Z(S) is a normal abelian subgroup of S/Z(S), then Z/Z(S) = Z(C_{G/Z(S)}(A/Z(S))) is a definable normal abelian subgroup of S/Z(S), and Z is a definable normal nilpotent subgroup of S. But R(S) is contained in Z(G) ∩ S, so we have Z^{∘} ≤ Z(G). Hence Corollary 4.8 implies that

[S, A] ≤ [S, Z] = [S, Z^{∘}] ≤ [S, Z(G)] = 1

and A is central in S. Thus S/Z(S) has no non-trivial normal abelian subgroup.

We prove that G = R(G)S and that R(G) ∩ S is central in S. Since G/Z(G) = R(G/Z(G))S_{0}/Z(G), we have G = R(G)S_{0} = R(G)SZ(G), and since G is definably connected and R(G) contains Z(G)^{∘}, we obtain G = R(G)S. Moreover, (R(G) ∩ S)Z(S)/Z(S) is a normal solvable subgroup of S/Z(S). Thus, since the previous paragraph says that S/Z(S) has no non-trivial normal abelian subgroup, R(G) ∩ S is contained in Z(S). □

Corollary 6.7.. – Let ℛ be an o-minimal expansion of a real closed field, and let G be an ℛ-definably connected ℛ-definable group. Then, for any subgroup S of G, the following conditions are equivalent:

• S is a maximal ind-definable semisimple subgroup (in the sense of [6]);

• S is the derived subgroup of a maximal quasi-semisimple subgroup.

Proof – Let S be a maximal ind-definable semisimple subgroup of G. By Fact 6.3 (2), there is a maximal semisimple subgroup T/Z(G)^{∘} of G/Z(G)^{∘} such that S = T′. Since Z(G)/Z(G)^{∘} is finite, T Z(G)/Z(G) is a maximal semisimple subgroup of G/Z(G). But the maximal semisimple subgroups of G/Z(G) are conjugate by Facts 5.1 and 6.3, so Theorem 6.6 provides a maximal quasi-semisimple subgroup L of G such that T Z(G)/Z(G) = LZ(G)/Z(G). Hence we have

S = T′ = (T Z(G))′ = (LZ(G))′ = L′

Now the result follows from the conjugacy of the maximal ind-definable semisimple subgroups in G (Fact 6.3) and from the conjugacy of the maximal quasi-semisimple subgroups in G (Theorem 6.6). □

References

[1] Elías Baro, Eric Jaligot, and Margarita Otero. Commutators in groups definable in o-minimal structures. Proc. Amer. Math. Soc., 140(10):3629–3643, 2012.

[2] Alexandre Borovik and Ali Nesin. Groups of finite Morley rank, volume 26 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications.

[3] Jeffrey Burdges. A signalizer functor theorem for groups of finite Morley rank. J. Algebra, 274(1):215–229, 2004.

[4] Gregory Cherlin. Good tori in groups of finite Morley rank. J. Group Theory, 8(5):613–621, 2005.

[5] Annalisa Conversano. On the connections between definable groups in o-minimal structures and real Lie groups: the non-compact case. Ph. D. thesis, University of Siena. 2009.

[6] Annalisa Conversano and Anand Pillay. On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures. Fund. Math., 222(1):49–62, 2013.

[7] Jamshid Derakhshan and Frank O. Wagner. Nilpotency in groups with chain conditions. Quart. J. Math. Oxford Ser. (2), 48(192):453–466, 1997.

[8] Martyn R. Dixon. Sylow theory, formations and Fitting classes in locally finite groups, volume 2 of Series in Algebra. World Scientific Publishing Co., Inc., River Edge, NJ, 1994.

[9] Klaus Doerk and Trevor Hawkes. Finite soluble groups, volume 4 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1992.

[10] Mario J. Edmundo. Structure theorems for o-minimal expansions of groups. Ann. Pure Appl. Logic, 102(1-2):159–181, 2000.

[11] Mário J. Edmundo. Solvable groups definable in o-minimal structures. J. Pure Appl. Algebra, 185(1-3):103–145, 2003.

[12] Pantelis E. Eleftheriou, Ya’acov Peterzil, and Janak Ramakrishnan. Interpretable groups are definable. J. Math. Log., 14(1):1450002, 47, 2014.

[13] Olivier Frécon. Étude des groupes résolubles de rang de Morley fini. Ph. D. thesis, Université de Lyon. 2000.

[14] Olivier Frécon. Around unipotence in groups of finite Morley rank. J. Group Theory, 9(3):341– 359, 2006.

[15] Olivier Frécon. Conjugacy of Carter subgroups in groups of finite Morley rank. J. Math. Log., 8(1):41–92, 2008.

[16] Olivier Frécon. Groupes géométriques de rang de Morley fini. J. Inst. Math. Jussieu, 7(4):751– 792, 2008.

[17] Olivier Frécon. Pseudo-tori and subtame groups of finite Morley rank. J. Group Theory, 12(2):305–315, 2009.

[18] Chris Miller and Sergei Starchenko. A growth dichotomy for o-minimal expansions of ordered groups. Trans. Amer. Math. Soc., 350(9):3505–3521, 1998.

[19] Margarita Otero. A survey on groups definable in o-minimal structures. In Model theory with applications to algebra and analysis. Vol. 2, volume 350 of London Math. Soc. Lecture Note Ser., pages 177–206. Cambridge Univ. Press, Cambridge, 2008.

[20] Margarita Otero, Ya’acov Peterzil, and Anand Pillay. On groups and rings definable in ominimal expansions of real closed fields. Bull. London Math. Soc., 28(1):7–14, 1996.

[21] Y. Peterzil, A. Pillay, and S. Starchenko. Definably simple groups in o-minimal structures. Trans. Amer. Math. Soc., 352(10):4397–4419, 2000.

[22] Y. Peterzil, A. Pillay, and S. Starchenko. Linear groups definable in o-minimal structures. J. Algebra, 247(1):1–23, 2002.

[23] Ya’acov Peterzil and Sergei Starchenko. Definable homomorphisms of abelian groups in ominimal structures. Ann. Pure Appl. Logic, 101(1):1–27, 2000.

[24] Ya’acov Peterzil and Charles Steinhorn. Definable compactness and definable subgroups of o-minimal groups. J. London Math. Soc. (2), 59(3):769–786, 1999.

[25] Anand Pillay. On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra, 53(3):239–255, 1988.

[26] Anand Pillay and Charles Steinhorn. Definable sets in ordered structures. I. Trans. Amer. Math. Soc., 295(2):565–592, 1986.

[27] Lou van den Dries. Tame topology and o-minimal structures, volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1998.

[28] Robert B. Warfield, Jr. Nilpotent groups. Lecture Notes in Mathematics, Vol. 513. Springer-Verlag, Berlin-New York, 1976.

Laboratoire de Mathématiques et Applications, Université de Poitiers

E-mail address: olivier.frecon@math.univ-poitiers.fr