Not-so-small representations of SL_{2} in the finite Morley rank category

Gregory Cherlin and Adrien Deloro

November 11, 2009

In this article we consider representations of SL_{2} which are interpretable in finite Morley rank theories, meaning that inside a universe of finite Morley rank we shall study the following definable objects: a group G isomorphic to SL_{2}, an abelian group V, and an action of G on V; V is thus a definable G-module on which G acts definably. Our goal will be to identify V with a standard Gmodule, under an assumption on its Morley rank. (A word on this notion of rank will be said shortly, after we have stated the results.)

It will be convenient to work with a faithful representation, possibly replacing SL_{2} by the quotient PSL_{2}, and we shall write G ≃ (P)SL_{2} to cover both cases.

Theorem. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a faithful action of G on V for which V is G-minimal. Assume rk V ≤ 3 rk 𝕂. Then V bears a structure of 𝕂-vector space such that:

• either V ≃ 𝕂^{2} is the natural module for G ≃ SL_{2}(𝕂), or

• V ≃ 𝕂^{3} is the irreducible 3-dimensional representation of G ≃ PSL_{2}(𝕂) with char 𝕂 ≠ 2.

On the way we shall establish the following interesting results.

Proposition 2.3. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a non-trivial action of G on V. Then for v generic in V, C_{G}^{∘}(v) is semi-simple or unipotent (possibly trivial).

Proposition 2.5. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂 of characteristic 0, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a faithful action of G on V for which V is G-minimal. Assume that for v generic in V, C_{G}^{∘}(v) is a non-trivial unipotent subgroup. Then V is a natural module for G. In particular G ≃ SL_{2}(𝕂).

All statements above involve the Morley rank of a structure; the reader should bear in mind that it is an abstract analog of the Zariski dimension, which can be axiomatized by some natural properties [2]. The Morley rank is however not related a priori to any geometry or topology, being a purely model-theoretic notion. Yet in general if a field 𝕂 has Morley rank k and V is an algebraic variety of (Zariski-)dimension d over 𝕂, then its Morley rank is dk. The rank hypothesis in the Theorem would thus amount, if the configuration were known to be algebraic, to assuming that dim V ≤ dim G; but of course the possibility for a field to have a finite Morley rank k > 1 makes algebraic geometry less general than our frame.

The latter is a ranked universe in the sense of [2]. Indeed, the semi-direct product V ⋊G is a ranked group in the sense of Borovik and Poizat [7, Corollaire 2.14 and Théorème 2.15]. In a word, we shall not go too deep into purely modeltheoretic arguments but will merely use the natural, intuitive properties, which can be taken as the definition of the Morley rank.

Let us now say a word about the proof of the Theorem. As we have mentioned, there is no geometry a priori on V ⋊G, and our efforts will be devoted to retrieving a suitable vector space structure on V which arises from the action of G. Model-theoretically speaking, the main tool is Zilber’s so-called Field Theorem [7, Théorème 3.7], which enables one to find an (algebraically closed) field inside a solvable, non-nilpotent, infinite group of finite Morley rank. A major difficulty is that there is no immediate reason why the action of an algebraic torus of G should induce such a structure on all of V. And even if a good structure exists, this does not mean that G itself is linear on V. The 2-dimensional case relies on a theorem by Timmesfeld (Fact 1.1 below); as for dimension 3, we rather manually extend the field action and some miraculous computations will, in the end, prove linearity of G.

Now that we have said what the present paper is, let us say what it is not: it does not relate directly to the classification project for simple groups of finite Morley rank.

1 Preparatory Remarks

We shall use throughout a characterization of the natural module which is due to Timmesfeld.

Fact 1.1 ([9, Theorem 3.4]). Let 𝕂 be a field and let G ≃ (P)SL_{2}(𝕂). Let V be a faithful G-module. Suppose the following:

(i). C_{V} (G) = 0

(ii). [U, U, V] = 1, where U is a maximal unipotent subgroup of G.

Let 0 ≠ v ∈ C_{V} (U) and W = ⟨v^{G}⟩. Then there exists a field action of 𝕂 on W such that W is the natural G-module. In particular G ≃ SL_{2}(𝕂).

We shall use the non-standard notation (+) to denote quasi-direct sum, i.e. the sum of two submodules (of a fixed module) which have a finite, possibly non-trivial, intersection.

1.1 On Malcev’s Theorem

Fact 1.2 ([7, Théorème 3.18]). Let G be a connected, solvable group of finite Morley rank acting definably and faithfully on a definable, abelian group A. If a definable subgroup B ≤ A is G- or G′-minimal, then B is centralized by G′.

Lemma 1.3. In a universe of finite Morley rank, consider the following definbale objects: a reductive algebraic group G, a nilpotent group V, and an action of G on V. Let U be a unipotent subgroup of G. Then V ⋊ U is nilpotent.

Proof. We may assume that U is a maximal unipotent subgroup. In this case, and by reductivity of G, U is the commutator subgroup of the Borel subgroup B = N_{G}(U) [1, p. 65]. Now consider H = V ⋊ B and write F^{∘}(H) = V ⋊ K. Clearly K ≤ F^{∘}(B), and H/F^{∘}(H) ≃ B/K is abelian by [2, Theorem 9.21], so U = B′ ≤ K.

Corollary 1.4. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a quasi-simple algebraic group G over 𝕂, an abelian group V, and a non-trivial action of G for which V is G-minimal. Then V has the same characteristic as 𝕂.

Proof. Let U be a maximal unipotent subgroup of G. By Lemma 1.3, V ⋊ U is nilpotent. If p = 0 and V is torsion or if p ≠ 0 and pV = V, then Burdges’ structure theorem for nilpotent groups [3, Theorem 2.31] yields [V, U] = 0. As conjugates of U generate G, the action is trivial, a contradiction.

1.2 Algebraicity in characteristic 0

We specialize [6] to our context.

Fact 1.5 (special case of [6, Theorem 4]). In a universe of finite Morley rank, consider the following definable objects: an abelian, torsion-free group A, an infinite group S, and a faithful action of S on A for which A is S-minimal. Then there is a subgroup A_{1} ≤ A and a field 𝕂 such that A_{1} ≃ 𝕂_{+} definably, and S embeds into GL_{n}(𝕂) for some n.

Lemma 1.6. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a simple algebraic group G over 𝕂, a torsion-free abelian group V, and a faithful action of G on V for which V is G-minimal. Then V ⋊ G is algebraic. Moreover, any definable subgroup of V has rank a multiple of rk 𝕂.

Proof. The assumptions imply that G is interpretable in 𝕂 as a pure field. By Fact 1.5, there is a field structure 𝕃 such that V ≃ 𝕃_{+}^{n} and G ↪ GL_{n}(𝕃) definably. 𝕃 has of course characteristic 0. By a result of Macpherson and Pillay (see [8, Theorem 3]), G is Zariski-closed in GL_{n}(𝕃) ; so far G and V ⋊ G are algebraic groups over 𝕃. In particular G as a pure group interprets 𝕃, so 𝕂 as a pure field interprets 𝕃. It follows that 𝕂 ≃ 𝕃 by [7, Théorème 4.15].

Now consider a definable subgroup V_{1} of V. Then the setwise stabilizer of V_{1} in 𝕂 is a definable, non-trivial subgroup of 𝕂, whence equal to 𝕂 by [7, Corollaire 3.3]. Hence V_{1} is a vector space on 𝕂, which proves that its rank is a multiple of rk 𝕂.

As a consequence, one can drastically simplify certain identification results in characteristic 0. For example, the following simplification of part of [5] results.

Theorem 1.7 ([5, Theorem A in char. 0]). Let G be a connected, non-solvable group of finite Morley rank acting definably and faithfully on a torsion-free connected abelian group V of Morley rank 2. Then there is an algebraically closed field 𝕂 of Morley rank 1 and characteristic 0 such that V ≃ 𝕂_{+}^{2}, and G is isomorphic to GL_{2}(𝕂) or SL_{2}(𝕂) in its natural action.

Proof. V is clearly G-minimal. By Fact 1.5, there is an interpretable field structure 𝕂 such that G ↪ GL_{n}(𝕂) with V ≃ 𝕂^{n}. Clearly the dimension must be 2, making the rank of the field 1. So there is a field 𝕂 of rank 1 such that V ≃ 𝕂_{+}^{2} and G ↪ GL_{2}(𝕂). But definable subgroups of GL_{2}(𝕂), especially over a field of rank 1, are known: [8, Theorem 5] together with connectedness and non-solvability of G forces either G ≃ GL(V) or G ≃ SL(V).

1.3 A Theorem of Frank Wagner

Fact 1.8 ([10, Corollary 9]). Let 𝕂 be a field of finite Morley rank of characteristic p > 0. Then 𝕂^{×} has no definable torsion-free section.

A good torus is a definable, abelian, divisible group with no torsion-free definable section; the latter condition being equivalent to: every definable subgroup is the definable hull of its torsion subgroup. Wagner’s Theorem states that in finite Morley rank, the multiplicative group of a field of characteristic p is a good torus.

Lemma 1.9. In a universe of finite Morley rank, consider the following definable objects: two infinite, abelian groups K and H, and a faithful action of K on H for which H is K-minimal. Suppose that H has exponent p and that K contains a non-trivial q-torus for each q ≠ p. Then rk H = rk K.

Proof. By Zilber’s Field Theorem, there is a field structure 𝕃 such that K ↪ 𝕃^{×} and H ≃ 𝕃_{+}. In particular, char 𝕃 = p. Now 𝕃^{×}/K is torsion-free, so by Wagner’s Theorem, K cannot be proper in 𝕃^{×}. Hence rk K = rk 𝕃 = rk H.

Lemma 1.10. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂 of characteristic p, a connected subgroup Θ of 𝕂^{×}, an abelian group V, and an action of Θ on V. Then there is θ ∈ Tor Θ such that C_{V} (Θ) = C_{V} (θ) and [V, Θ] = [V, θ].

Proof. By Wagner’s Theorem, 𝕂^{×} is a good torus, hence Θ = d(Tor Θ). By the descending chain condition on centralizers, C_{V} (Θ) = C_{V} (Tor Θ) = C_{V} (θ_{1}, …, θ_{n}) for torsion elements, and we take a generator θ_{0} of the finite cyclic group ⟨θ_{1}, …, θ_{n}⟩: one has C_{V} (Θ) = C_{V} (θ_{0}), and this holds true of any root of θ_{0}.

Now the group [V, Tor Θ] is definable, so Σ = {t ∈ Θ : [V, t] ≤ [V, Tor Θ]} is a definable subgroup of Θ containing Tor Θ. Again, as Θ = d(Tor Θ), it follows Σ = Θ, that is [V, Θ] = [V, Tor Θ]. By the ascending chain condition on the lattice of connected groups [V, t] (t ∈ Tor Θ), there is θ ∈ Tor Θ such that [V, θ] = [V, Tor Θ] = [V, Θ]. We may assume that θ is a root of θ_{0}, and we are done.

1.4 Cohomological computations

Fact 1.11. Let A be a connected, abelian group of finite Morley rank of bounded exponent and α a definable automorphism of finite order coprime to the exponent of A. Then A = C_{A}(α) ⊕ [A, α]. Moreover, if A_{0} < A is a definable, connected, α-invariant subgroup, then [A, α] ∩ A_{0} = [A_{0}, α].

Proof. Let ad_{α} and Tr_{α} be the adjoint and trace maps, that is

ad_{α}(x) = x^{α} − x and Tr_{α}(x) = x + ⋯ + x^{α^{n−1}}

where n is the order of α. It is easily seen, as A has no n-torsion, that ker ad_{α} ∩ ker Tr_{α} = 0. In particular, rk A ≥ rk (ker ad_{α}) + rk (ker Tr_{α}). Moreover, im ad_{α} ≤ ker Tr_{α} and im Tr_{α} ≤ ker ad_{α}. It follows therefore that rk A ≥ rk (ker ad_{α}) + rk (ker Tr_{α}) ≥ rk (ker ad_{α}) + rk (im ad_{α}) = rk A, so im ad_{α} = ker Tr_{α}. Hence A = ker ad_{α} + ker Tr_{α} = ker ad_{α} ⊕ im ad_{α} = C_{A}(α) ⊕ [A, α].

Let a_{0} ∈ A_{0}; then a_{0} ∈ ad_{α}(A_{0}) iff Tr_{α}(a_{0}) = 0 iff a_{0} ∈ ad_{α}(A).

Corollary 1.12. Let A be a connected elementary abelian p-group of finite Morley rank and T a p^{⊥} good torus acting definably on A. Then A = C_{A}(T) ⊕ [A, T]. Let A_{0} < A be a definable, connected, T -invariant subgroup. Then C_{A}(T) covers C_{A/A_{0}}(T). Moreover, C_{T} (A) = C_{T} (A_{0}, A/A_{0}).

Proof. Since T is a good torus we may work as in Lemma 1.10 to find a torsion element t ∈ T such that C_{A}(T) = C_{A}(t) and [A, T] = [A, t]. We apply Fact 1.11 and deduce that A = C_{T} (A) ⊕ [A, T].

If x ∈ A maps to an element in C_{A/A_{0}}(t), then denoting the canonical projection by π one has π ad_{t}(x) = ad_{t} π(x) = 0. Hence ad_{t}(x) ∈ A_{0} and by Fact 1.11 there is x_{0} ∈ A_{0} such that ad_{t}(x) = ad_{t}(x_{0}), whence x ∈ x_{0} + ker ad_{t}, and ker ad_{t} = C_{A}(t).

Now let Θ = C_{T} (A_{0}, A/A_{0}); as T is a good torus, Θ is the definable hull of its torsion subgroup. Let t ∈ Tor Θ. Then as above C_{A}(t) covers C_{A/A_{0}}(t) = A/A_{0}; it follows that A = C_{A}(t) + A_{0} ≤ C_{A}(t). Hence t ∈ C_{T} (A) and Θ = d(Tor Θ) ≤ C_{T} (A).

1.5 Automorphisms of semi-direct products

Lemma 1.13. In a universe of finite Morley rank, let A, T be definable, abelian, infinite groups such that A is T -minimal and the action is faithful. Let K be a definable group normalizing A and T. Then K centralizes T.

Proof. We let K act on End A by:

s^{φ}(a) := (s(a^{φ^{−}1}))^{φ}

But by assumption, K normalizes the image of T in End A, which additively generates an algebraically closed field (this is the proof of Zilber’s field theorem). In particular, as there are no definable groups of automorphisms of a field of finite Morley rank [2, Theorem 8.3], K acts trivially on T.

2 Actions of (P)SL_{2}

The present section is devoted to general actions of (P)SL_{2} in the finite Morley rank category, with no assumption on the rank itself. Proposition 2.3 is our main result. The following notations will be adopted in §§2 and 3.

Notation 2.1. Let G ≃ (P)SL_{2}. Fix a Borel subgroup B of G and let U = B′ be its unipotent subgroup. Let T be an algebraic torus such that B = U ⋊ T. Let i be the involution in T, and ζ ∈ N(T) a 2-element inverting T (the order of ζ depends on the isomorphism type of G).

Let us start with a classical observation.

Lemma 2.2. A definable, connected subgroup of (P)SL_{2} is semi-simple, unipotent, or contains a maximal unipotent subgroup of (P)SL_{2}.

Proof. Let K be a definable, connected subgroup. We may assume that K is proper; as K is then solvable (see for instance [8, Théorème 4]), up to conjugacy K ≤ B. It follows that K = U_{1} ⋊ T_{1} with obvious notations. If U_{1}, T_{1} > 1 then T_{1} additively generates 𝕂, so U_{1} must be stable under 𝕂^{×}, and U_{1} = U.

2.1 Actions of (P)SL_{2} and centralizers

Proposition 2.3. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a non-trivial action of G on V. Then for v generic in V, C_{G}^{∘}(v) is semi-simple or unipotent (possibly trivial).

Proof. Let V be as in the statement; we first show that we may assume C_{V} (G) = 0. Assume the result holds when C_{V} (G) = 0 and let V as in the statement. Let V_{0} = C_{V} (G) < V. Since G is perfect, one has C_{V/V_{0}}(G) = 0, and the action of G on V/V_{0} is non-trivial. By assumption, the result holds for V/V_{0}. Now let v ∈ V be generic. Then v̄ ∈ V/V_{0} is generic too, and in particular C_{G}^{∘}(v̄) is either semi-simple or unipotent. As C_{G}^{∘}(v) ≤ C_{G}^{∘}(v̄), we are done.

So from now on we suppose C_{V} (G) = 0. In Notation 2.1 we had fixed a maximal unipotent subgroup U ≤ G; B = N(U) its normalizer; T an algebraic torus such that B = U ⋊ T; and a 2-element ζ inverting T.

Let v ∈ V be generic. C_{G}^{∘}(v) is proper in (P)SL_{2}, hence solvable [8, Théorème 4]; up to conjugacy, C_{G}^{∘}(v) ≤ B. Assume that C_{G}^{∘}(v) is neither unipotent nor semi-simple. Then by Lemma 2.2, C_{G}^{∘}(v) contains U.

So up to conjugacy, C_{G}^{∘}(v) = U ⋊T_{v} for some non-trivial T_{v} ≤ T. The family {C_{G}^{∘}(v)/U : v ∈ V : U ≤ C_{G}^{∘}(v) ≤ B} of subgroups of T is uniformly definable; as T ≃ 𝕂^{×} is a good torus, the family is finite [4, Rigidity II]. It follows that there is a common T_{0} ≤ T such that generically, C_{G}^{∘}(v) is conjugate to U ⋊ T_{0}.

Now let V_{1} = C_{V} (U). Clearly V_{1} is infinite, taking a B-minimal subgroup of V and applying Malcev’s Theorem (Fact 1.2). As any two distinct conjugates of U generate G and C_{V} (G) = 0, V_{1} must be disjoint from V_{1}^{g} for g ∉ B. It follows that N_{G}(V_{1}) = B and that V_{1} is disjoint from its distinct conjugates. One therefore has

rk V_{1}^{G} = rk V_{1} + rk G − rk B = rk V_{1} + rk 𝕂.

Notice that by assumption, the generic element of V is centralized by a conjugate of U ⋊ T_{0}. Thus V_{1}^{G} is generic in V. But furthermore, for v generic in V_{1}, C_{G}^{∘}(v) is a conjugate of U ⋊ T_{0} containing U; conjugacy is therefore obtained by an element of N(U) = B. As B′ = U, U ⋊ T_{0} is normal in B; hence C_{G}^{∘}(v) = U ⋊ T_{0}. This means that T_{0} centralizes a generic subset X of V_{1}; as X + X = V_{1} it follows that V_{1} = C_{V} (U ⋊ T_{0}).

Let W = V_{1} ⊕ V_{1}^{ζ} and W̌ = W \ (V_{1} ∪ V_{1}^{ζ}). The generic element of W is in W̌. Let v ∈ W̌. Clearly T_{0} ≤ C_{G}^{∘}(v). Moreover, if C_{G}^{∘}(v) is not semisimple, then it must meet a unipotent subgroup which can only be either U or U^{ζ} as 1 ≠ T_{0} ≤ C_{G}^{∘}(v). In that case, C_{G}^{∘}(v) contains either U or U^{ζ} by Lemma 2.2, against the definition of W̌. This means that for v ∈ W̌, one has T_{0} ≤ C_{G}^{∘}(v) ≤ T. In particular, W̌^{G} is not generic in V.

It follows that W < V. As V_{1}^{G} is generic in V, W cannot be G-invariant. Therefore T ·⟨ζ⟩ ≤ N_{G}(W) < G, and equality follows from maximality of T ·⟨ζ⟩. As T · ⟨ζ⟩ also normalizes V_{1} ∪ V_{1}^{ζ}, one sees that N_{G}(W̌) = T · ⟨ζ⟩.

Let w ∈ W̌. Assume that w ∈ W̌^{g} for some g ∈ G. Then C_{G}^{∘}(v) is a nontrivial subgroup of T, so C_{G}(C_{G}^{∘}(v)) = T = T^{g}, and g ∈ N(T) = T ·⟨ζ⟩ = N(W̌). This implies that

rk W̌^{G} = rk W̌ + rk G − rk T = 2 rk V_{1} + 2 rk 𝕂 = 2 rk V_{1}^{G}.

But V_{1}^{G} is already generic in V which is infinite: this is a contradiction.

Corollary 2.4. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a non-trivial action of G on V. Then rk V ≥ 2 rk 𝕂.

2.2 The characteristic 0 case

We continue in the vein of Proposition 2.3.

Proposition 2.5. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂 of characteristic 0, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a faithful action of G on V for which V is G-minimal. Assume that for v generic in V, C_{G}^{∘}(v) is a non-trivial unipotent subgroup. Then V is a natural module for G. In particular G ≃ SL_{2}(𝕂).

Proof. Since V is torsion-free and G-minimal, C_{V} (G) = 0. Let Ḡ = G/C_{G}(V); Ḡ is still isomorphic to (P)SL_{2}(𝕂) and now faithful on V. We apply Lemma 1.6 and retrieve a 𝕂-vector space structure on V: in particular every definable subgroup has rank a multiple of rk 𝕂.

Let v ∈ V be generic; as there are no non-trivial, proper, definable subgroups of 𝕂_{+} [7, Corollaire 3.3], C_{G}^{∘}(v) is a conjugate of a maximal unipotent subgroup.

Let V_{1} = C_{V} (U) < V; as any two distinct conjugates of U generate G, V_{1} is disjoint from V_{1}^{g} for g ∉ B. Thus N_{G}(V_{1}) = B and V_{1} meets its distinct conjugates trivially. By our assumption, V_{1}^{G} is generic in V. So

rk V = rk V_{1}^{G} = rk V_{1} + rk G − rk N_{G}(V_{1}) = rk V_{1} + rk 𝕂.

In particular rk V/V_{1} = rk 𝕂. Now V_{1} has rank a multiple of rk 𝕂, so rk V_{1} ≥ rk 𝕂. It follows that V_{1} ⊕ V_{1}^{ζ} = V, and V/V_{1} ≃ V_{1}^{ζ} ≃ 𝕂_{+} has no proper non-trivial definable subgroups.

Thus B acts on V/V_{1} which is B-minimal, and by Malcev’s Theorem (Fact 1.2) B′ = U centralizes V/V_{1}. So [V, U, U] = 1 and we may apply Timmesfeld’s identification result, Fact 1.1. It follows that V is the natural module for Ḡ ≃ SL_{2}(𝕂), and this forces G ≄ PSL_{2}(𝕂) but G ≃ SL_{2}(𝕂) and G = Ḡ.

Unfortunately, nothing similar seems to be possible in characteristic p.

2.3 Four-groups of PSL_{2}

We finish this section with an easy but useful relation on ranks when the characteristic is not 2. Given a definable, involutive automorphism j of an abelian group W of finite Morley rank with no involutions, one has W = W^{+_{j}} ⊕ W^{−_{j}} with obvious notations [2, Exercise 14 p.73].

Lemma 2.6. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ PSL_{2}(𝕂), an abelian group V of characteristic not 2, and a non-trivial action of G on V. Then

rk V = rk (V^{+_{i}+_{ζ}}) + \frac{3}{2} rk (V^{−_{i}})

Proof. Write V = V^{+_{i}} ⊕V^{−_{i}}, then V^{+_{i}} = V^{+_{i}+_{ζ}} ⊕V^{+_{i}−_{ζ}} and V^{−_{i}} = V^{−_{i}+_{ζ}} ⊕ V^{−_{i}−_{ζ}}. Let a = rk V^{+_{i}+_{ζ}} and b = rk V^{+_{i}−_{ζ}}. Clearly, b = rk V^{−_{i}+_{ζ}} = rk V^{−_{i}−_{ζ}}. It follows rk V^{−_{i}} = 2b and rk V = a + 3b.

This lemma will yield a crucial estimate of ranks in Corollary 3.15.

3 Proof of the Theorem

We now attack our main result.

Theorem. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a faithful action of G on V for which V is G-minimal. Assume rk V ≤ 3 rk 𝕂. Then V bears a structure of 𝕂-vector space such that:

• either V ≃ 𝕂^{2} is the natural module for G ≃ SL_{2}(𝕂), or

• V ≃ 𝕂^{3} is the irreducible 3-dimensional representation of G ≃ PSL_{2}(𝕂) with char 𝕂 ≠ 2.

Notation 3.1. In a universe of finite Morley rank, consider the following definable objects: a field 𝕂, a group G ≃ (P)SL_{2}(𝕂), an abelian group V, and a non-trivial action of G on V for which V is G-minimal. Assume rk V ≤ 3 rk 𝕂.

One should also bear in mind Notation 2.1 which introduces the usual elements and subgroups of (P)SL_{2}.

Notation 3.2. Let k = rk 𝕂 and write rk V = 2k + ν.

Notice that 0 ≤ ν ≤ k by Corollary 2.4 and our assumption that rk V ≤ rk G. Moreover, if ν = 0, then by [5, Theorem B] (which is a consequence, in finite Morley rank, of Timmesfeld’s identification result, Fact 1.1), we are done. So we suppose ν > 0 throughout. Our goal is to show that the characteristic is not 2, ν = k, and G ≃ PSL_{2} acts on V ≃ 𝕂^{3} in the usual irreducible way.

If V has characteristic 0, then by Lemma 1.6, V ⋊ G or V ⋊ G/Z(G) is algebraic; dim_{𝕂} V is 2 or 3, and as irreducible algebraic representations of (P)SL_{2} are well-known, the analysis already ends. From now on, we suppose char 𝕂 to be a prime number p. The proof will involve studying various submodules of V, defining an action piecewise, and eventually proving its linearity. On our way we shall prove p ≠ 2, though a more direct attack could be possible.

Lemma 3.3. We may suppose that C_{V} (G) = 0.

Proof. Suppose our Theorem holds for modules with a trivial right-kernel. Notice that by G-minimality, W = C_{V} (G) is finite. It follows that there is no right kernel for G on the G-minimal module V̄ = V/W; so the result holds for the action of G on V̄. In particular, as we have assumed rk V > 2k, we find that char 𝕂 ≠ 2 and G ≃ PSL_{2}(𝕂), so that ⟨i, ζ⟩ is a four-group. We also know that ζ inverts a set of rank 2k in V̄.

It follows that ζ inverts a set of rank ≥ 2k in V. Hence rk V^{−_{ζ}} ≥ 2k, and Lemma 2.6 implies that V^{+_{i}+_{ζ}} is finite. As it is clearly connected, we deduce that W ≤ V^{+_{i}+_{ζ}} = 0.

Bear in mind on the contrary that when char 𝕂 = 2, SL_{2}(𝕂) acts on sl_{2}(𝕂) by conjugacy with an infinite right-kernel. Characteristic 2 will be eliminated in Lemma 3.13 below.

3.1 T -invariant sections

Our finer study of submodules starts here. A word on terminology: if K is a group acting on an abelian group V, we shall call any definable, connected, K-invariant subgroup a K-submodule.

In this subsection only, we work with abstract T -modules (of finite Morley rank) which need not relate to our current representation V.

Definition 3.4. Call a T -module X degenerate if C_{T}^{∘} (X) ≠ 1.

We now consider cork C_{T}^{o} (X) = rk(T/C_{T}^{o} (X)).

Lemma 3.5.

(i). Let X be a T -module. Then cork C_{T}^{∘} (X) ≤ rk X.

If X is degenerate, then cork C_{T}^{∘} (X) < rk X.

(ii). Let X be a T · ⟨ζ⟩-module. Then cork C_{T}^{∘} (X) ≤ \frac{rk X}{2} .

If X is degenerate, then cork C_{T}^{∘} (X) < \frac{rk X}{2} .

Proof. Let Θ = C_{T}^{∘} (X).

(i). Let 0 = X_{0} < X_{1} < ⋯ < X_{n} = X be a maximal series of T -modules, and Θ_{i} = C_{T}^{∘} (X_{i}/X_{i−1}). As Θ = (∩_{i}Θ_{i})^{∘}, one has cork Θ ≤ ∑_{i} cork Θ_{i}. So we may assume that X itself is T -minimal.

By Zilber’s Field Theorem, there is a field structure 𝕃 such that T/Θ embeds into 𝕃^{×} and X ≃ 𝕃_{+}; the first claim follows. If in addition X is degenerate, that is if we know Θ ≠ 1, then by Wagner’s Theorem Θ must contain torsion; as Θ is connected it follows that T/Θ ≄ 𝕃^{×}, and the embedding is proper, whence the second claim.

(ii). Considering a maximal series of T · ⟨ζ⟩-modules, we may now assume that X is T · ⟨ζ⟩-minimal.

Let Y ≤ X be a T -minimal T -submodule. If Y < X, then Y ∩ Y^{ζ} is finite, and X = Y (+)Y^{ζ}. Moreover C_{T}^{∘} (Y) = Θ. Applying (i) we find cork Θ ≤ rk Y = \frac{rk X}{2} , the inequality being strict if X is degenerate.

We now suppose that Y = X, that is X is T -minimal. But now Lemma 1.13 forces the action of ζ to be trivial on T/Θ, whence Θ = T, and the claim is obvious.

3.2 Maximin

Proposition 3.6. The largest degenerate T -submodule of V exists; it has rank ≤ ν.

Proof.

Step 1. There is a non-trivial degenerate T -submodule of V.

Proof: Suppose not. Let V_{1} ≤ V_{2} ≤ V be B-submodules, with V_{1} and V_{2}/V_{1} B-minimal. Notice that by Malcev’s Theorem (Fact 1.2), both V_{1} and V_{2}/V_{1} are even T -minimal. Notice further that V_{2} < V, as otherwise the action is quadratic, and Fact 1.1 yields a contradiction.

If rk V_{1} ≠ k then by Lemma 1.9 T_{1} = C_{T}^{∘} (V_{1}) must be infinite; taking C_{V}^{∘} (T_{1}) ≥ V_{1} we are done. So we may assume rk V_{1} = k.

Suppose rk V_{2}/V_{1} ≠ k. As V_{2}/V_{1} is T -minimal, the group T_{2} = C_{T}^{∘} (V_{2}/V_{1}) is non-trivial by Lemma 1.9. Let t_{2} ∈ Tor T_{2} be given by Lemma 1.10 (with respect to the action on V); V_{2} = C_{V_{2}}(t_{2})⊕[V_{2}, t_{2}]. Now C_{V_{2}}(T_{2}) = C_{V_{2}}(t_{2}) covers V_{2}/V_{1} by Corollary 1.12, so C_{V_{2}}(T_{2}) is non-trivial; in particular C_{V}^{∘} (T_{2}) ≠ 1: we are done.

So suppose rk V_{2}/V_{1} = k, that is rk V_{2} = 2k, and let W_{2} = (V_{2} ∩ V_{2}^{ζ})^{∘}. Clearly rk W_{2} ≥ 2k − ν > 0. If (V_{1} ∩ W_{2})^{∘} ≠ 0, then by T -minimality of V_{1}, one has V_{1} ≤ W_{2}. By T -minimality of V_{1} and V_{2}/V_{1}, one finds that W_{2} is either V_{1} or V_{2}, a contradiction as neither is ζ-invariant.

Therefore (V_{1} ∩ W_{2})^{∘} = 0, and in particular V_{2} = V_{1}(+)W_{2}; whence W_{2} is T -minimal, and ζ-invariant. As ζ inverts T, Lemma 1.13 then forces T to centralize W_{2}: we are done. ♢

Step 2. Any degenerate T -submodule of V has rank ≤ ν.

Proof: Let X be degenerate and Θ = C_{T}^{∘} (X) ≠ 1. We first claim that for x generic in X, C_{G}^{∘}(x) is semi-simple. Otherwise, as C_{G}^{∘}(x) contains Θ ≤ T, it contains either U or U^{ζ}; we may assume that for x generic in X, U centralizes x. Thus U centralizes X. As the latter is ζ-invariant, it follows that G = ⟨U, U^{ζ}⟩ centralizes X, a contradiction.

Hence, the centralizer in G of the generic element of X is semi-simple. Let x ∈ X be generic, and suppose that there is g ∈ G such that x ∈ X^{g}. Then C_{G}^{∘}(x) ≥ ⟨Θ, Θ^{g}⟩ which is semi-simple. Then C_{G}^{∘}(⟨Θ, Θ^{g}⟩) is an algebraic torus, which can be only C_{G}^{∘}(Θ) = T, and only T^{g} for a similar reason. Hence g ∈ N_{G}(T) = T · ⟨ζ⟩ = N_{G}(X). So X is generically disjoint from its distinct conjugates; it follows that

rk X^{G} = 2k + rk X ≤ rk V = 2k + ν

which proves the claim. ♢

Step 3. The sum of two degenerate T -submodules is degenerate.

Proof: Let X_{1}, X_{2} be degenerate T -submodules, and Θ_{i} = C_{T}^{∘} (X_{i}) ≠ 1. Considering X̂_{i} = C_{V}^{∘} (Θ_{i}) ≥ X_{i}, we may assume that the X_{i} are T · ⟨ζ⟩-modules.

By Lemma 3.5 (ii) cork_{T} Θ_{i} < \frac{rk X_{i}}{2} , so using Step 2 rk Θ_{i} > k − \frac{ν}{2} ≥ \frac{k}{2} . It follows that Θ_{12} = (Θ_{1} ∩ Θ_{2})^{∘} is non-trivial. Now X_{12} = C_{V}^{∘} (Θ_{12}) contains X_{1} + X_{2}. ♢

This concludes the proof of Proposition 3.6.

Notation 3.7. Let X be the largest degenerate T -submodule of V, and let Θ = C_{T}^{∘} (X). Let θ_{0} ∈ Tor Θ be given by Lemma 1.10 (for the action on V), so that X = C_{V} (θ_{0}) and [V, Θ] = [V, θ_{0}].

Lemma 3.8. U does not centralize X.

Proof. Assume it does. Then since X is ζ-invariant, so does U^{ζ}. It follows that ⟨U, U^{ζ}⟩ = G centralizes X ≠ 0, a contradiction.

We shall eventually prove that T = Θ centralizes X (Proposition 3.14 below). But let us first introduce some useful objects.

3.3 Minimax and a commutator subgroup

Notation 3.9. Let M = [V, Θ] = [V, θ_{0}] (see Notation 3.7).

One has V = M ⊕ X by Corollary 1.12.

Lemma 3.10. M has rank 2k. Moreover, non-trivial proper T -submodules of M are T -minimal and have rank k.

Proof. As V = C_{V} (θ_{0}) ⊕ M and X = C_{V} (θ_{0}), we know cork M ≤ ν by Proposition 3.6, so rk M ≥ 2k. Let V_{1} ≤ M be a T -minimal T -submodule. If rk V_{1} ≠ k then by Lemma 1.9, C_{T}^{∘} (V_{1}) is infinite, and it follows from Proposition 3.6 that V_{1} ≤ X by definition of X, a contradiction to X ∩ M = 0. Hence V_{1} has rank k.

Now let V_{1} < V_{2} ≤ M be a T -submodule such that V_{2}/V_{1} is T -minimal. If rk V_{2}/V_{1} ≠ k, then by Lemma 1.9 again, T_{2} = C_{T}^{∘} (V_{2}/V_{1}) is infinite. C_{V_{2}}(T_{2}) covers V_{2}/V_{1}, but by Proposition 3.6 it lies in X. Since X ∩ M = 0, we have a contradiction.

So rk V_{2} = 2k. If V_{2} < M, then there is V_{3} ≤ M such that V_{3}/V_{2} is T - minimal. But M < V, so rk V_{3}/V_{2} < k, and we argue as before to deduce that V_{3} ≤ V_{2} +X, whence V_{3} = V_{2}, a contradiction. Hence M = V_{2} has rank 2k. It is now clear that any non-trivial proper T -submodule of M is actually T -minimal, and therefore has rank k.

Notation 3.11. Let Y = [X, U].

In the representation of PSL_{2} in its Lie algebra, this should capture the set of upper-triangular matrices.

Lemma 3.12. Y is a subgroup of M of rank k. It is B-minimal and C_{T} (Y) ≤ Z(G). Moreover, M = Y ⊕ Y^{ζ}; U centralizes Y, (X + Y)/Y, and V/(X + Y).

Proof. Let u ∈ U, x ∈ X, and θ ∈ Θ. We denote by · the action of T on U (isomorphic to the action of 𝕂^{×} on 𝕂_{+}). One has

[[x, u], θ]] = [x, u]^{θ} − [x, u]
= [x, θ · u] − [x, u]
= [x, (θ − 1) · u]

So [x, (θ − 1) · u] ∈ [V, θ] ≤ M. But (Θ − 1) · U = U, ,so [x, U] ≤ M and Y = [X, U] ≤ M. We claim that Y is neither 0 nor M. By Lemma 3.12, Y > 0, and Y is clearly definable, connected, and T -invariant. If Y = M, then Y is ⟨U, ζ⟩ = G-invariant, which contradicts G-minimality of V. So 0 < Y < M.

By Lemma 3.10, Y is T -minimal and has rank k. Now by construction, U normalizes Y; so does T, hence Y is B-invariant. In particular, Y is B-minimal. By Lemma 1.3 [Y, U] < Y, so [Y, U] = 0, that is U centralizes Y. If Y were ζ-invariant, we would have that G = ⟨U, U^{ζ}⟩ centralizes Y, a contradiction. By T -minimality and rank estimates, it follows that M = Y + Y^{ζ}; if there is a non-trivial intersection then T must centralize this intersection, against Y ∩ X ≤ M ∩ X = 0. So M = Y ⊕ Y^{ζ}.

We now consider C_{T} (Y). Suppose that there exists t ∈ C_{T} (Y) which is not central in G. Let M̌ = M \ (Y ∪ Y^{ζ}); t centralizes M and M̌. Notice that N_{G}(M̌) = T · ⟨ζ⟩. Now let g ∉ N_{G}(M̌). If there is m ∈ M̌ ∩ M̌^{g}, then C_{G}(m) contains t and t^{g}, so it can’t be semi-simple; it must therefore contain a maximal unipotent subgroup, which can only be U or U^{ζ}. So one has for instance U ≤ C_{G}(m), which proves m ∈ C_{M}(U) = Y by Lemma 3.10; this violates m ∉ Y and the other case is similar. So distinct conjugates of M̌ are disjoint, which blows up the rank of V. Hence C_{T} (Y) ≤ Z(G).

It remains to show that U centralizes V/(X + Y). As X + Y is B-invariant, V/(X + Y) is a B-module. Now V/(X + Y) ≃ M/Y as a T -module, so V/(X + Y) is T -minimal, whence B-minimal. Therefore U centralizes V/(X + Y) by Malcev’s Theorem, Fact 1.2.

By Zilber’s Field Theorem, there is a field structure 𝕃_{1} such that Y ≃ (𝕃_{1})_{+} and T/C_{T} (Y) ↪ 𝕃_{1}^{×}. But C_{T} (Y) is finite, so T/C_{T} (Y) ≃ 𝕂^{×}; by Wagner’s Theorem it follows that T/C_{T} (Y) ≃ 𝕃_{1}^{×}. Hence Y is a 𝕂-vector space.

We eventually get rid of the characteristic 2 case.

Lemma 3.13. p ≠ 2.

Proof. Suppose p = 2. We show first that [V, U, U] = 0.

Fix u ∈ U^{#}. Then since Θ additively generates 𝕂, we have C_{X}(u) = C_{X}(u, Θ) = C_{X}(U) = 0. Since Y ≤ C_{V} (U) we find C_{(X+Y)}(u) = Y + C_{X}(u) = C_{(X+Y)}(U). Now [V, u] ≤ X + Y by Lemma 3.12; also, [V, u] ≤ C_{V} (u) since p = 2. So [V, u] ≤ C_{X+Y} (u) ≤ C_{V} (U). Hence [V, U, U] = 0.

Since C_{V} (G) = 0 (Lemma 3.3) and C_{G}(V) = 1, Fact 1.1 forces V to contain an isomorphic copy W of the natural representation. Using indecomposability, W must clearly be definable; as it is G-invariant, it follows V = W has rank 2k, a contradiction.

From now on the characteristic is an odd prime p. This will play a crucial role in the rank computation of Corollary 3.15. But we shall now make a first incursion into linear structures.

3.4 Finer study of X

As promised, we prove the following.

Proposition 3.14. Θ = T centralizes X.

Proof. If [T, X] ≤ C_{X}(U), then since [T, X] is ζ-invariant one even has [T, X] ≤ C_{X}(U, U^{ζ}) ≤ C_{V} (G) = 0: we are done.

So we assume that X̄ = ([T, X] + C_{X}(U))/C_{X}(U) is not 0 and shall prove a contradiction.

Notation 1.

• Let W be a T -minimal T -subgroup of X̄. (Notice that T does not centralize W, since Corollary 1.12 shows that CX̄(T) = 0.)

• Let 𝕃 be the definable field structure such that W ≃ 𝕃_{+} and T/C_{T} (W) ↪ 𝕃^{×}. We use ∗ to denote multiplication in 𝕃.

There is such a structure indeed, as we have observed that T does not centralize W. We shall embed 𝕃 into 𝕂 definably. This will eventually force rk 𝕃 = rk 𝕂, which will be very close to a conclusion. Let us work on constructing such a definable field embedding.

Fix u_{0} ∈ U^{#}, w_{0} ∈ W^{#}. Recall that W is made of classes modulo C_{X}(U), so [w_{0}, u_{0}] denotes a well-defined element of Y. Moreover, this element is nontrivial, as otherwise ⟨Θ · u_{0}⟩ = U centralizes w_{0}, and w_{0} = 0 in W. So [w_{0}, u_{0}] is a well-defined, non-trivial element of Y ≃ 𝕂_{+}. We introduce the following operation.

Notation 2. Fix u_{0} ∈ U^{#} and w_{0} ∈ W^{#}. For w′ ∈ W^{#} there is a unique k ∈ 𝕂^{×} such that

[w′, u_{0}] = k · [w_{0}, u_{0}]

Step 3. k depends on w_{0} and w′, but not on u_{0}.

Proof: Since T ≃ 𝕂^{×}, there is t ∈ T such that k · [w_{0}, u_{0}] = [w_{0}, u_{0}]^{t}. Let u′ be any element of U^{#}. As Θ additively generates 𝕂, there are elements θ_{1}, …, θ_{n} ∈ Θ such that u′ = θ_{1} · u_{0} + ⋯ + θ_{n} · u_{0}. Since Θ centralizes W, it follows that:

[w′, u′] = ∑_{i}[w′, θ_{i} · u_{0}]  = ∑_{i}[w′, u_{0}]^{θ_{i}}
= ∑_{i}(k · [w_{0}, u_{0}])^{θ_{i}} = ∑_{i}[w_{0}, u_{0}]^{tθ_{i}}
= ∑_{i}[w_{0}, u_{0}]^{θ_{i}t}  = (∑_{i}[w_{0}, u_{0}]^{θ_{i}})^{t}
= k · ∑_{i}[w_{0}, u_{0}]^{θ_{i}} = k · [w_{0}, ∑_{i} θ_{i} · u_{0}] = k · [w_{0}, u′]

which proves the claim. ♢

Notation 4. For ℓ ∈ 𝕃, let f(ℓ) ∈ 𝕂 be defined as the unique k such that:

[ℓ ∗ w_{0}, u_{0}] = k · [w_{0}, u_{0}]

Step 5. f is a definable field embedding.

Proof: f is clearly additive. We now show that it is multiplicative. Let ℓ_{1}, ℓ_{2} ∈ 𝕃; let k_{1}, k_{2} be their images through f. First assume that ℓ_{1} and ℓ_{2} are in the image of T/C_{T} (W): there exist elements t_{1}, t_{2} ∈ T which map to ℓ_{1}, ℓ_{2}. Then bearing in mind that f does not depend on the base point u_{0} (Step 3), one has:

f(ℓ_{1} ∗ ℓ_{2}) · [w_{0}, u_{0}] = [(ℓ_{1} ∗ ℓ_{2}) ∗ w_{0}, u_{0}] = [w_{0}^{t_{1}t_{2}}, u_{0}]
= [w_{0}^{t_{1}}, u_{0}^{t_{2}^{−1}}]^{t_{2}}  = [ℓ_{1} ∗ w_{0}, u_{0}^{t_{2}^{−1}}]^{t_{2}}
= (k_{1} · [w_{0}, u_{0}^{t_{2}^{−1}}])^{t_{2}} = [w_{0}, u_{0}^{t_{2}^{−1}}]^{t_{1}t_{2}}
= [w_{0}, u_{0}^{t_{2}^{−1}}]^{t_{2}t_{1}}  = k_{1} · ([w_{0}, u_{0}^{t_{2}^{−1}}])^{t_{2}}
= k_{1} · [w_{0}^{t_{2}}, u_{0}]  = k_{1} · [ℓ_{2} ∗ w_{0}, u_{0}]
= k_{1} · k_{2} · [w_{0}, u_{0}]  = (f(ℓ_{1}) · f(ℓ_{2})) · [w_{0}, u_{0}]

It follows that f(ℓ_{1}∗ℓ_{2}) = f(ℓ_{1})·f(ℓ_{2}). By additivity, f is actually multiplicative on the set additively generated by the image of T in 𝕃, that is on all of 𝕃. ♢

Now we have a definable field embedding from 𝕃 to 𝕂. As there are no extensions of infinite fields in a universe of finite Morley rank, this forces

rk 𝕃 = rk 𝕂 = rk Y = k ≥ ν ≥ rk X ≥ rk W = rk 𝕃

In particular ν = k, and W is a quotient of X by a finite subgroup; so X is T -minimal. As it is ζ-invariant and ζ inverts T, Lemma 1.13 implies Θ = T.

This is a contradiction, and in particular T does centralize X. All our construction collapses: the field structure 𝕃 vanishes.

Proposition 3.14 is now proved. Remember that we have lost the field stucture 𝕃 on which the argument relied.

3.5 Consequences

Before moving to the identification argument we draw important consequences of Proposition 3.14.

Corollary 3.15. G ≃ PSL_{2} and ζ inverts X; ν = k is the rank of X; C_{V} (U) = Y is disjoint from X.

Proof. By Proposition 3.14 T centralizes X, so the involution i of T cannot invert V; we may suppose G ≃ PSL_{2}. As G ≃ PSL_{2}(𝕂), there is no central element; by Lemma 3.12, T is faithful on Y. In particular, i inverts Y and Y^{ζ}, so it inverts M = Y ⊕ Y^{ζ}. It follows that rk V^{−_{i}} ≥ 2k, and by Lemma 2.6, one finds ν = k. Moreover V^{+_{i}+_{ζ}} = 0 and this means that ζ inverts X. Also, since V = M ⊕X (see Notation 3.9) and rk M = 2k (Lemma 3.10), one has rk X = k.

We now let Ŷ = C_{V} (U); if Y < Ŷ, let Y ≤ Z ≤ Ŷ be such that Z/Y is B-minimal. As Ŷ is disjoint from Ŷ^{ζ}, it has rank at most \frac{3k}{2} , so rk Z/Y < k. By Lemma 1.9, C_{T} (Z/Y) ≠ 1; Z contains a non-trivial degenerate module, that is C_{X}(U) ≠ 0. As ζ inverts X, C_{X}(U) is also centralized by U^{ζ} while C_{V} (G) = 0. Hence Ŷ = Y and X ∩ Y = 0.

This is indeed the case in the three-dimensional representation of PSL_{2}. We now work towards understanding the scalar action on X. Corollary 3.16. Let x ∈ X, t ∈ T, u ∈ U^{#}. Then there is a unique x′ ∈ X such that [x′, u] = [x, u]^{t} = [x, t · u]; x′ depends on x and t, but not on u. Proof. Fix u_{1} ∈ U^{#} and consider the definable morphism from X to Y which maps x to [x, u_{1}]. This is injective, as the kernel lies in C_{X}(u_{1}) = C_{X}(T, u_{1}) ≤ C_{X}(U) = 0. By equality of ranks, the map is a bijection. Now suppose another u_{2} ∈ U^{#} is given, and we have elements x′_{1}, x′_{2} such that [x′_{i}, u_{i}] = [x, u_{i}]^{t}. Then there is τ ∈ T such that u_{2} = u_{1}^{τ}, and it follows that:

\begin{aligned}[x′_{2}, u_{2}] & = & [x, u_{2}]^{t} & = & [x, u_{1}^{τ}]^{t} & = & [x, u_{1}]^{τt} \\ & = & [x, u_{1}]^{tτ} & = & [x′_{1}, u_{1}]^{τ} & = & [x′_{1}, u_{1}^{τ}] & = & [x′_{1}, u_{2}] \end{aligned}

whence x′_{1} = x′_{2}, as claimed.

3.6 Identification

Let us serve some refreshments.

• ζ has order 2 (Corollary 3.15)

• V = Y ⊕ X ⊕ Y^{ζ} (Notation 3.9 and Lemma 3.12).

• Y = [X, U] is B-minimal (Lemma 3.12)

• X = C_{V} (T) is inverted by ζ (Corollary 3.15)

• X and Y have rank k.

We define a 𝕂-scalar action on each component: Notation 3.17.

• On Y, k · y is given by the action of T.

• On Y^{ζ}, we let k · y^{ζ} = (k · y)^{ζ}.

• On X, we let k · x be the unique x′ ∈ X such that [x′, u] = k · [x, u] (Corollary 3.16; this does not depend on the choice of u).

We shall check that G acts linearly. We do it piecewise; notice that when we claim that U acts linearly on X, we mean that the operation induced by elements of U from X to V is linear, without claiming anything about invariance under the action. Lemma 3.18. T · ⟨ζ⟩ acts linearly on V. U acts linearly on Y ⊕ X.

Proof. By construction, T is linear on Y and Y^{ζ}. It is linear on X, as it acts trivially! By construction, ζ is linear on Y ⊕Y^{ζ}. As it inverts X, it is also linear on X. So T · ⟨ζ⟩ is linear on V.

As U acts trivially on Y, it is linear on Y. It remains to see that U is linear on X. Let u ∈ U, x ∈ X, and k ∈ 𝕂. By definition of the action on X, one has [k · x, u] = k · [x, u], and therefore:

k · x^{u} − k · x = k · [x, u] = [k · x, u] = (k · x)^{u} − k · x

Linearity follows.

It remains to prove that U is linear on Y^{ζ}. As T is, and since it acts transitively on U^{#}, it suffices to exhibit one non-trivial element of U which is linear on Y^{ζ}.

Notation 3.19 (Bryant Park element). Let w = ζ (it is an involution, after all). Let u ∈ U be such that (wu) has order 3.

Such an element exists (this may be viewed as a special case of the Steinberg relations). We shall prove that this particular u is linear on Y^{ζ}.

Lemma 3.20. For any y ∈ Y, there is a unique x ∈ X such that y^{wu} = y + x + y^{w}.

Proof. A priori, one has

y^{wu} = y_{1} + x + y_{2}^{w}

for elements y_{1}, y_{2} ∈ Y and x ∈ X. But U centralizes Y, (X + Y)/Y, and V/(X + Y) by Lemma 3.12. So y_{2} = y. We push further, using the fact that w inverts X (Corollary 3.15).

\begin{aligned}y^{(wu)^{2}} & = & y_{1}^{wu} + x^{wu} + y^{wwu} \\ & = & y_{1}^{wu} − x^{u} + y \end{aligned}

and

y = y^{(wu)^{3}} = y_{1}^{wuwu} − x^{uwu} + y^{wu}

whence applying u^{−1},

y = y_{1}^{wuw} − x^{uw} + y^{w}

Now U^{w} centralizes Y^{w}, X +Y^{w}/Y^{w}, and V/(X +Y^{w}) (Lemma 3.12 again), so [u^{w}, y_{1}] ∈ X + Y^{w}. It follows that y_{1} is the projection on Y of y_{1}^{wuw}. On the other hand, x^{u} ∈ X + Y, so x^{uw} ∈ X + Y^{w}. Taking projections on Y modulo X + Y^{w}, one has y_{1} = y.

Lemma 3.21. Let y ∈ Y and x ∈ X be as in Lemma 3.20. Then [x, u] = 2y.

Proof. By definition,

y^{wu} = y + x + y^{w}

Let us iterate:

y^{(wu)^{2}} = y^{wu} + x^{wu} + y^{wwu}
= (y + x + y^{w}) − x^{u} + y
= 2y + x − x^{u} + y^{w}

and

y^{(wu)^{3}} = 2y^{wu} + x^{wu} − x^{uwu} + y^{wwu}
= 2(y + x + y^{w}) − x^{u} − x^{uwu} + y
= 3y + 2x − x^{u} − x^{uwu} + 2y^{w}

As wu has order three, one has:

2y + 2x − x^{u} − x^{uwu} + 2y^{w} = 0

Now u centralizes (Y + X)/Y, so there is y_{1} ∈ Y such that x^{u} = x + y_{1}. Let x_{1} be associated to y_{1} by Lemma 3.20: one has y_{1}^{wu} = y_{1} + x_{1} + y_{1}^{w}. Hence

x^{uwu} = x^{wu} + y_{1}^{wu}
= −x^{u} + (y_{1} + x_{1} + y_{1}^{w})
= −x − y_{1} + y_{1} + x_{1} + y_{1}^{w}
= x_{1} − x + y_{1}^{w}

It follows that

2y + 2x − (x + y_{1}) − (x_{1} − x + y_{1}^{w}) + 2y^{w} = 0,

and projecting onto Y modulo X + Y^{w},

y_{1} = 2y

so that [x, u] = y_{1} = 2y. Notation 3.22. For y ∈ Y, let x(y) be the element x given by Lemma 3.20. Lemma 3.23. The function x(y) is 𝕂-linear. Proof. Let k ∈ 𝕂. Then

[x(k · y), u] = 2(k · y) = k · (2y) = k · [x(y), u] = [k · x(y), u]

And we are done. Corollary 3.24. u is linear on Y^{w}. Proof. Let y ∈ Y and k ∈ 𝕂; let y_{2} = k · y, and x_{2} = x(y_{2}). Then

(k · y^{w})^{u} = y_{2}^{wu} = y_{2} + x_{2} + y_{2}^{w} = k · y + x_{2} + k · y^{w}

On the other hand,

k · y^{wu} = k · (y + x + y^{w}) = k · y + k · x + k · y^{w}

As x is 𝕂-linear, both expressions are equal: u is linear on Y^{w}.

It follows that G = ⟨T, ζ, u⟩ is linear on V. We may now finish the proof. For clarity let us denote by 𝕂′ the isomorphic copy of 𝕂 which is definable in V. We have a definable embedding ι of G ≃ PSL_{2}(𝕂′) into GL_{3}(𝕂′). The algebraic torus T of G maps to an algebraic torus of GL_{3}(𝕂′); hence ι(T) is Zariski-closed in GL_{3}(𝕂′). Now ι(G) = ⟨ι(T^{g}) : g ∈ G⟩; it follows that ι(G) is Zariski-closed in GL_{3}(𝕂′). So G, which is definably isomorphic to ι(G), is algebraic over 𝕂′. This concludes the identification together with the proof of our Theorem.

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