Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 121–133.

Solvable Collineation Groups of Smooth Projective Planes

Richard Bödi

Mathematisches Institut, Universität Tübingen D - 72076 Tübingen, Germany

Abstract. Let Δ be a closed solvable subgroup of the collineation group of one of the four classical Moufang planes 𝒫_{2}𝔽 over the domain 𝔽 ∈ {ℝ, ℂ, ℍ, 𝕆}, where ℍ denotes the quaternions and 𝕆 the Cayley numbers. Then dim Δ ≤ 4, 9, 17, 30 if 𝔽 = ℝ, ℂ, ℍ, 𝕆. For arbitrary compact connected projective planes 𝒫 of finite dimension n, M. Lüneburg derived upper bounds for the dimension of Δ depending on n and the configuration of the fixed elements of Δ, see [17]. In this paper we consider dim Δ in the case of smooth projective planes. Compared to the results of [17], the upper bounds of dim Δ can be lowered in almost every case; except for one single case we even derive sharp bounds.

0. Introduction

Stable planes originate from Salzmann’s paper Topological planes [24] and have been studied systematically by R. Löwen and by M. Stroppel, see for example [14], [15], [26], and [27]. We started the investigation of smooth stable planes in [1]. In this paper we continue the study of collineation groups of a smooth stable plane 𝒮 (see [2] and [3]). This time we will restrict our attention to closed solvable subgroups Δ of the group Γ = Aut(𝒮) of continuous collineations of 𝒮. As in [3], we want to derive upper bounds for the dimensions of Δ. Our proofs rely on the fact that up to duality a solvable collineation group of a compact connected topological projective plane has a global fixed point p, i.e. Δ = Δ_{p} holds. This, of course, is not valid for stable planes, since one can remove the fixed point p thus obtaining a smooth stable subplane on which Δ acts without any fixed points. Hence, we consider smooth projective planes instead of smooth stable planes.

1. Definitions and auxiliary results

Throughout we will use the notation and the definitions of [1] and [2].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1998 Heldermann Verlag Definition 1.1. A projective plane 𝒫 = (P, ℒ) is called smooth if the set P of points and the set ℒ of lines are smooth (= C^{∞}) manifolds such that the geometric operations ∨ of join and ∧ of intersection are smooth mappings.

It is convenient to identify every line L ∈ ℒ with the set of points incident with L. If we look at a line this way, we will refer to L as a point row. Dually, the set ℒ_{p} of all lines through some point p is called a line pencil. The integer n = dim P which is called the dimension of the projective plane 𝒫 is always a power 2^{k} of 2, where k = 1, 2, 3, 4, see [25], §54. Moreover, for n = 2l we have dim L = dim ℒ_{p} = l for any point row L and any line pencil ℒ_{p}.

By [1], the tangent spaces T_{p}P (p ∈ P) and T_{L}ℒ (L ∈ ℒ) together with the tangent spreads 𝒮_{p} = {T_{p}K | K ∈ ℒ_{p}} and 𝒮_{L} = {T_{L}ℒ_{q} | q ∈ L} induced by the line pencil ℒ_{p} and the point row L are locally compact affine translation planes 𝒜_{p} and 𝒜_{L}, respectively. Their projective closures are denoted by 𝒫_{p} and 𝒫_{L} and in both cases we put L_{∞} to be the line at infinity (which is also the translation line). According to [2], (2.3), every continuous collineation of 𝒫 is in fact smooth. This enables us to compare the results in the topological situation with the results in the smooth case, see the end of this paper. The group Γ of continuous (or, equivalently, of smooth) collineations γ of 𝒫 is a Lie transformation group (with respect to the compact-open topology) on both the set P of points and the set ℒ of lines, see [2], (2.4). In particular, any collineation γ is a diffeomorphism of P onto P and of ℒ onto ℒ. The stabilizer Γ_{p} of some point p ∈ P induces an action on the tangent translation plane 𝒜_{p} via the derivation mapping

D_{p} : Γ_{p} → Σ_{0} := Aut(𝒜_{p})_{0} : γ ↦ Dγ(p).

By [2], (3.3), the map D_{p} is a continuous homomorphism and ker D_{p} = Γ_{[p,p]} is the subgroup of all elations of Γ having p as their center.

We use the small inductive dimension ind rather than the (more common) covering dimension dim because of its local nature. In most cases, however, both of these notions of dimension coincide, for instance this is true for locally compact groups and for manifolds, of course. Thus, we will write dim even if we think of the small inductive dimension. Without further mentioning we will always use Halder’s dimension formula, [12] and also [25], 96.10, as well as the following lemma.

Lemma 1.2. If Δ is a locally compact connected collineation group of a smooth projective plane 𝒫 which fixes some element x ∈ P ∪ ℒ, then

dim Δ = dim ker D_{x} + dim D_{x}Δ and dim D_{x}Δ ≤ dim Aut(𝒜_{x})_{0}.

Proof. By Theorem (3.3) of [2], the group Δ acts on the Lie group Σ_{0} = Aut(𝒜_{x})_{0} via composition (Δ, γ) ↦ D_{x}(Δ)γ. By Halder [12], this implies that

dim Δ/Δ_{1} = dim 1^{Δ} = dim D_{x}Δ ≤ dim Σ_{0},

where the last inequality follows from the monotonicity of the small inductive dimension. From Δ_{1} = ker D_{x} we conclude that

dim Δ/ker D_{x} = dim D_{x}Δ,

and hence dim Δ = dim ker D_{x} + dim D_{x}Δ follows from Nagami [20], (2.1). □

Proposition 1.3. Let 𝒫 be a compact connected translation plane of dimension 2^{k}, where 1 ≤ k ≤ 4, and let L_{∞} be some translation axis of 𝒫. Let K, L ≠ L_{∞} be two different lines such that o = K ∧L does not lie on L_{∞}. If Δ is a closed connected solvable subgroup of the stabilizer ∇ of the triangle defined by K, L, L_{∞}, then dim Δ ≤ 2 + rk Δ, where rk Δ ≤ k is the dimension of a maximal torus subgroup of Δ. In the case k = 1 we have rk Δ = 0.

Proof. By H. Hähl [9], (4.2), the connected component ∇^{1} of ∇ possesses a maximal compact subgroup K of codimension at most 2. Since a compact connected solvable Lie group is isomorphic to some torus group 𝕋^{r}, we infer that a closed solvable subgroup Δ of ∇^{1} has dimension at most r + 2. According to M. Lüneburg, [17], II, Satz 2 (see also [25], 55.37), we have r ≤ k. If k = 1, then 𝒫 is isomorphic to the projective plane over the reals (see H. Salzmann, [24], (7.24)), and hence rk Δ = 0 follows, because in this case Δ is a subgroup of the group of all regular real diagonal matrices. □

As usual, we write Γ_{[c]} for the group of all central collineations with center c, and we write Γ_{[c,A]} for the subgroup of Γ_{[c]} consisting of all collineations with center c and axis A. Moreover, the group of all elations with center c is denoted by Γ_{[c,c]}. The following result deals with the structure of groups of central collineations. The second part of the next proposition is also true in the case of compact connected projective planes (see Hähl, [10]). Nevertheless we will give a (simpler) proof which uses the special situation of smoothness.

Proposition 1.4. Let 𝒫 be a smooth projective plane and let p ∈ P. Let Δ ≠ 1l be a connected locally compact collineation group of central collineations of 𝒫 with center p. Then the derivation mapping D_{p} maps Δ onto a closed subgroup Υ of Aut(𝒫_{p}), and Δ = Δ_{[p,A]} ⋉ Δ_{[p,p]} ≅ Υ ⋉ Δ_{[p,p]} for some line A ∉ ℒ_{p}.

Proof. By [2], (4.12), the image Υ = D_{p}Δ is a closed (connected) subgroup of ℍ^{×}. If Υ ≠ 1l, then Υ contains a nontrivial central element ω. If Δ_{[p,p]} = ker D_{p} = 1l, then D_{p} maps Δ isomorphically (as a topological group) onto Υ and Δ consists entirely of homologies. Moreover, the collineation δ := D_{p}^{−1}(ω) is a homology of 𝒫 with some line A as its axis. Because δ is central in Δ, we infer that δ fixes every line of the orbit A^{Δ}. From Δ = Δ_{[p]} we infer that the orbit A^{Δ} does not contain any line incident with the center p. This implies that A^{Δ} = A, whence A is the axis of every homology of Δ, i.e. we have Δ = Δ_{[p,A]}. Now let Δ_{[p,p]} ≠ 1l, and assume that Δ ≠ Δ_{[p,p]} (otherwise there is nothing left to prove). Fix some collineation δ in the inverse image D\overleftarrow{p}(ω). Then δ is a homology, since D_{p}(δ) ≠ 1l. Again, let A be the axis of δ. Because the point p does not lie on the line A, we infer from Δ_{[p,p]} ≠ 1l that the line A is moved by Δ. By Hähl, [10], dual theorem of (1.3), this implies that A^{Δ} = A^{Δ_{[p,p]}}. Now let γ ∈ Υ and choose ε ∈ D\overleftarrow{p}(γ).

Then there is some (uniquely determined) elation τ ∈ Δ_{[p,p]} such that A^{ε} = A^{τ}, whence we have ετ^{−1} ∈ Δ_{[p,A]}. Thus we have proved that for every γ ∈ Υ there exists exactly one collineation in ε ∈ D\overleftarrow{p}(γ) ∩ Δ_{[p,A]}. In particular, the restriction D_{p} : Δ_{[p,A]} → Υ is a (closed) isomorphism, i.e. the groups Δ_{[p,A]} and Υ are isomorphic as topological groups. This proves the proposition. □

Corollary 1.5. Let 𝒫 be a smooth projective plane and let p ∈ P. Then the connected component of the group Δ_{[p]} is a linear group.

Proof. Let Υ denote the connected component of Δ_{[p]}. According to the last proposition, we can write Υ = Υ_{[p,A]} ⋉ Υ_{[p,p]}. The normal subgroup Υ_{[p,p]} is simply connected by [2], (4.16), and the complement Υ_{[p,A]} is a subgroup of the multiplicative group of the quaternion numbers by [2], (3.14). In particular, this complement is a reductive group. By [13], Chapter XVIII, Thereom 4.3, this implies the linearity of Υ. □

Remark. Similar results concerning the linearity of certain subgroups of Γ can be found in [2], especially Theorem (4.19) and Corollary (4.20).

Theorem 1.6. Let 𝒫 be a smooth projective plane of dimension 2^{k}, let K, L be two different lines, and put p = K ∧ L. If Δ is a closed connected solvable subgroup of the stabilizer of K and L, then

dim Δ ≤ dim Δ_{[p,p]} + rk Δ + 2.

Moreover, if one of the following conditions (i) dim D_{p}Δ ≥ rk Δ + 2,

(ii) 𝒜_{p} is classical and dim D_{p}Δ ≥ 3, 4, 6 (for k = 2, 3, 4)
(iii) k = 1 and dim D_{p}Δ ≥ 2,

is satisfied, then dim Δ_{[p]}/Δ_{[p,p]} ≥ 1 holds.

Proof. We consider the derivation map D_{p} : Δ → Σ_{0} = Aut(𝒜_{p})_{0}. Using Corollary (3.9) of [2], Proposition 1.3 together with Lemma 1.2 yields

dim Δ = dim ker D_{p} + dim D_{p}Δ ≤ dim Δ_{[p,p]} + rk Δ + 2.

From Hähl, [9], (4.2) or [25], 81.8, we know that D_{p}Δ is a subgroup of some group R·H_{k}·T_{k}, where R ≅ ℝ, H_{k} is a closed solvable subgroup of the homology group fixing the origin 0 and the line at infinity L_{∞}, T_{0} = 1l, and T_{k} ≅ 𝕋^{k} if k > 0, see also M. Lüneburg, [17], II, Satz 1. In particular, we infer that D_{p}Δ contains a one-parameter subgroup of homologies with center 0 if dim D_{p}Δ ≥ rk Δ + 2. Thus, Proposition 1.4 proves the second claim. Now we assume that 𝒜_{p} is a classical plane. By Salzmann [24], (7.24), this is always true for k = 1. Since D_{p}Δ fixes T_{p}L and T_{p}K, we infer that D_{p}Δ consists of diagonal real matrices if k = 1. Thus, we conclude from dim D_{p}Δ ≥ 2 that D_{p}Δ contains the homology group {λ · id | λ > 0} as a subgroup. Again, Proposition 1.4 shows that dim Δ_{[p]}/Δ_{[p,p]} ≥ 1. If k ∈ {2, 3}, then H_{k} ≅ ℂ^{×} and thus D_{p}Δ contains a one-parameter subgroup of homologies with center 0 if dim D_{p}Δ ≥ k + 1. Thus it remains to study the case k = 4. Since we have rk D_{p}Δ ≤ k = 4, this case is already covered by condition (i). Thus we have proved all parts of the theorem. □

Lemma 1.7. Let 𝒫 be a smooth projective plane of dimension 2l, and let p ∈ P and L ∈ ℒ_{p}. If t = dim Γ_{[p,L]}, then dim Γ_{[p,p]} ≤ l + t.

Proof. Choose some point x on L distinct from p, and select another point y not on L. Then the stabilizer (Γ_{[p,p]})_{x,y} is trivial and thus we have

dim Γ_{[p,p]} = dim x^{Γ_{[p,p]}} + dim y^{(Γ_{[p,p]})_{x}} ≤ l + t.  □

The last result of this section says that a solvable Lie group cannot act transitively on an n-sphere if n ≥ 2. For a proof see Montgomery and Samelson, [18], cp. also [25], 96.20, 96.23, and 96.28.

Lemma 1.8. Let G be a solvable Lie group acting on an n-sphere S. If n ≥ 2, then G cannot act transitively on S.

2. The dimensions of solvable groups

We are going to derive upper bounds for the dimensions of solvable groups Δ acting on a smooth projective plane 𝒫 = (P, ℒ) of finite positive dimension n = 2^{k} = 2l. By M. Lüneburg, [17], (G2), p. 22, the following four cases cover all possible situations which can occur for a closed connected solvable collineation group Δ of 𝒫.

(I) Δ fixes some flag (p, L) and dim Δ_{[p,L]} ≥ 1.

(II) Δ fixes some antiflag.

(III) Δ fixes some Baer subplane of 𝒫, i.e. a subplane of dimension l.

(IV) Every minimal connected commutative normal subgroup N of Δ has some orbit x^{N}, where x ∈ P ∩ ℒ, which generates a subplane of 𝒫.

The following theorems will treat each of these cases separately.

Theorem 2.1. Let 𝒫 be a smooth projective plane of dimension n = 2^{k} = 2l. Assume that the solvable group Δ fixes some flag (p, L) such that t := dim Δ_{[p,L]} ≥ 1 holds. Put d_{I}(1) = 3, d_{I}(2) = 8, d_{I}(3) = min {16, 13 + t}, and d_{I}(4) = min {29, 22 + t}. Then either dim Δ is at most d_{I}(k), or 𝒫 is isomorphic to the classical plane of dimension 2^{k}.

Proof. Let K ∈ ℒ_{p} \ {L}. If dim Δ_{[p,p]} = 2l, then 𝒫 is a dual translation plane, whence 𝒫 is a classical Moufang plane according to a theorem of J. Otte, [21] and [22]. Thus we may assume that dim Δ_{[p,p]} < 2l. Let D_{p} : Δ_{K} → Σ_{0} be the derivation mapping. By Lemma 1.7 and Corollary 1.6 we have

dim Δ = dim K^{Δ} + dim Δ_{K}

= dim K^{Δ} + dim ker D_{p} + dim D_{p}(Δ_{K})

≤ l + dim Δ_{[p,p]} + rk Δ + 2 ≤ l + (l + t) + rk Δ + 2

< 3l + k + 2 ≤ \{ \begin{matrix}5 & if k = 1 \\ 10 & if k = 2 \\ 17 & if k = 3 \\ 30 & if k = 4 \end{matrix}

(∗)

Furthermore, we obtain from inequality (∗)

dim Δ ≤ 2l + t + k + 2 ≤ \{ \begin{matrix}13 + t & if k = 3 \\ 22 + t & if k = 4 \end{matrix}

which proves the theorem for k ≥ 3. Now let us have a look at the cases k = 1, 2. Choose some point b ≠ p on L. Recalling that Δ fixes the point p, we get Δ_{[b,b]} = Δ_{[b,L]}. Thus

dim Δ = dim b^{Δ} + dim Δ_{b} ≤ l + dim ker D_{b} + dim D_{b}(Δ_{b}) ≤ l + dim Δ_{[b,L]} + dim D_{b}(Δ_{b})

holds.

Case k = 1. According to M. Lüneburg [17], III, Satz 1, there is a minimal connected closed abelian normal subgroup N of Δ with N = N_{[p,L]}. In particular, we have dim N = 1, and hence Δ is (p, L)-transitive (note that L \ {p} is connected). If dim Δ_{[b,L]} = 1, then L is a translation axis of 𝒫 and 𝒫 is classical by Salzmann, [24], (7.25). Thus we may assume that dim Δ_{[b,L]} = 0, whence dim Δ ≤ l + dim D_{b}(Δ_{b}). If we suppose that dim Δ = 4, the last inequality shows that dim D_{b}(Δ_{b}) ≥ 3. Since D_{b}(Δ_{b}) is a solvable subgroup of Σ_{0} ≤ GL_{2}ℝ, we infer that D_{b}(Δ_{b}) contains the group {λ · id | λ > 0} which consists of homologies (with center 0 and axis L_{∞}). By Lemma 1.4 the plane 𝒫 has a one-dimensional group H consisting of homologies with center b. Since H is commutative, there is a line K through p such that H = H_{[b,K]}, cp. [25], 61.25. Since dim Δ_{[p,p]} = 1 and dim Δ = 4, the same argument shows that there is a homology δ having p as its center and some line M as its axis. Since we may choose the point b arbitrarily on L \ {p}, we may assume that b does not lie on M ≠ L. Then (Δ_{[b,K]})^{δ} = Δ_{[b^{δ},K]} with b^{δ} ≠ b.

Applying [25], Lemma 33.3 to H and Δ_{[b^{δ},K]} ≠ 1l, we get a (p, K)-transitive translation group Δ_{[p,K]}. Consequently, we have dim Δ_{[p,p]} = 2, contrary to our assumptions. This shows that dim Δ ≤ 3 if 𝒫 is not isomorphic to 𝒫_{2}ℝ. Case k = 2. If dim Δ_{[p,p]} ≤ 2, then dim Δ ≤ 8 follows immediately by (∗). Hence we may suppose that dim Δ_{[p,p]} = 3 and thus Δ is (p, K)-transitive for some line K ∈ ℒ_{p}, see [25], dual statement of 61.11(c) together with 61.12. Assume moreover that dim Δ = 9. From

9 = dim Δ = dim ker D_{p} + dim D_{p}Δ = dim Δ_{[p,p]} + dim D_{p}Δ = 3 + dim D_{p}Δ

we conclude that dim D_{p}Δ = 6. Thus the tangent translation plane 𝒜_{p} has an automorphism group of dimension at least dim D_{p}Δ + dim T = 10, where T ≅ ℝ^{4} is the translation group of 𝒜_{p}. According to the classification of 4-dimensional compact translation planes (see [25], Chapt. 7) this implies that 𝒜_{p} is isomorphic to the affine complex plane 𝒜_{2}ℂ. In particular, we have D_{p}Δ ≤ GL_{2}ℂ. Since D_{p}Δ fixes the spread element T_{p}L, we have up to conjugacy

D_{p}Δ ≤ Υ := \{\binom{a b}{c} | a, b, c ∈ ℂ, ac ≠ 0\} ≤ GL_{2}ℂ

and from dim D_{p}Δ = 6 = dim Υ we conclude that D_{p}Δ = Υ, since both D_{p}Δ and Υ are connected. Thus

(D_{p}Δ)_{[p]} = \{\binom{a}{a} | a ∈ ℂ^{×}\}

is two-dimensional, and hence dim Δ_{[p]}/Δ_{[p,p]} ≥ 2. By Lemma 1.4 this implies that Δ contains a (commutative) two-dimensional group H consisting of homologies with center p and some axis M. Since K\{p, K ∧ M} is connected, this shows that H is (p, M)-transitive. Using the dual statement of Pickert, [23], 3.1.11, p. 67, we infer that 𝒫 is (K ∧ M, M)- transitive. Since p ≠ K ∧ M this implies that 𝒫 is (K, K)-transitive, see Pickert, [23], 3.1.12.1, p. 68. In particular, the plane 𝒫 is a translation plane. Hence 𝒫 is classical by Otte [22] or by T. Grundhöfer, H. Hähl, [8]. This finishes the proof of the theorem. □

Lemma 2.2. Let 𝒫 be a smooth projective planes of dimension n = 2l = 2^{k} with k ≥ 2. If the solvable group Δ fixes some point p, and if dim Δ_{[p,p]} = 0, then

dim Δ ≤ l + rk Δ + 2 ≤ l + k + 2 ≤ \{ \begin{matrix}6 & if k = 2 \\ 9 & if k = 3 \\ 14 & if k = 4 \end{matrix} .

Proof. By Lemma 1.8, the group Δ acts not transitively on ℒ_{p}. Hence either the group Δ fixes some line K ∈ ℒ_{p} or there is a non-trivial orbit K^{Δ} of dimension at most l − 1. If K^{Δ} = K, we may apply Corollary 1.6 in order to get

dim Δ = dim Δ_{K} = dim K^{′Δ_{K}} + dim Δ_{K,K′} ≤ l + k + 2

for any line K′ ∈ ℒ_{p} \ {K}. Now let K^{Δ} be a non-trivial orbit of dimension at most l − 1. Since Δ is solvable, there are normal subgroups P and Π of Δ with Π < P and K^{Π} = K ≠ K^{P} such that P/Π is a minimal normal subgroup of Δ/Π. Thus, the group Δ acts irreducibly on Υ := P/Π and Υ is isomorphic to one of the abelian groups 𝕋, ℝ, or ℝ^{2}. If dim Υ ≤ 1, then Δ induces on Υ a group Δ^{Υ} of dimension at most 1. If Υ ≅ ℝ^{2}, then the induced group Δ^{Υ} contains a torus subgroup for Δ acts irreducibly on Υ. This implies that dim Δ^{Υ} ≤ 2. Hence, in any case, the centralizer

Ξ = {ξ ∈ Δ | ∀ ρ ∈ P : Πρ^{ξ} = Πρ}

has codimension at most 2 in Δ and contains the group Π. For ξ ∈ Ξ with K^{ξ} = K we have K^{ρξ} = K^{ξ^{−1}ρξ} ∈ K^{Πρ} = K^{ρ}, whence Ξ_{K} acts trivially on the orbit K^{P}. In particular, the stabilizer Ξ_{K} fixes a second line K′ ∈ ℒ_{p}. Applying Proposition 1.3 and Corollary 1.6 to Ξ_{K} = Ξ_{K,K′}, we get

dim Δ ≤ dim Ξ + 2 ≤ dim K^{Ξ} + dim Ξ_{K} + 2 ≤ (l − 1) + (r + 2) + 2 ≤ l + k + 3.

If dim K^{Ξ} ≤ l −2 or if r ≤ k −1, the same computation shows that dim Δ ≤ l +k +2 holds, and the lemma is proved. Hence we may assume r = k as well as dim K^{Ξ} = l − 1. Since the kernel of the tangent translation plane 𝒜_{c} is isomorphic to some closed connected subgroup of the multiplicative group of the quaternions (see Buchanan, Hähl, [4]), and because r = k ≥ 2, there is a torus group T ≅ 𝕋 of Ξ_{K} that acts faithfully on ℒ_{p}. By E. Floyd, [5], Th. 5.2, the dimension of the fixed point set of an element δ ∈ T of order 3 is of codimension at least 2. In particular, the collineation δ acts non-trivially on K^{Ξ}, because the latter set has codimension 1. If we choose K^{∗} ∈ K^{Ξ} in such a way that K^{∗}^{δ} ≠ K^{∗}, then dim K^{∗}^{T} = 1. Consequently, the torus rank of Ξ_{K^{∗}} is at most r − 1 (recall that r is the torus rank of Ξ). Let ξ ∈ Ξ with K^{ξ} = K^{∗}. Since

Ξ_{K^{∗}} = Ξ_{K^{ξ}} = (Ξ_{K})^{ξ} = (Ξ_{K,K′})^{ξ} = Ξ_{K^{∗},K^{′ξ}},

we can apply Proposition 1.3 to Ξ_{K^{∗}} in order to obtain

dim Ξ_{K^{∗}} ≤ (r − 1) + 2 ≤ (k − 1) + 2 = k + 1.

This finally implies that

dim Δ ≤ dim Ξ + 2 ≤ dim K′^{Ξ} + dim Ξ_{K′} + 2 ≤ (l − 1) + (k + 1) + 2 = l + k + 2,

and the proof is completed. □

We turn to the antiflag case.

Theorem 2.3. Let 𝒫 be a smooth projective plane of dimension n = 2^{k} = 2l. Assume that the solvable group Δ fixes some antiflag (p, L). Put d_{II}(1) = 3, d_{II}(2) = 6, d_{II}(3) = 9, and d_{II}(4) = 14. Then the dimension of Δ is at most d_{II}(k).

Proof: Using Corollary 1.6 and noting that Δ_{[p,p]} = 1l we get

dim Δ = dim ker D_{p} + dim D_{p}Δ = dim Δ_{[p,p]} + dim D_{p}Δ = dim D_{p}Δ.

Case k = 1. Since D_{p}Δ is a solvable subgroup of Σ_{0} ≤ GL_{2}ℝ, we get dim D_{p}Δ ≤ 3. Case k ≥ 2. For k ≤ 3 we could give a similar proof as in the case k = 1 using the classification of 4-dimensional and 8-dimensional translation planes. However, the classification of 16-dimensional compact translation planes does not provide usable results for k = 4. Instead, we use Lemma 2.2 to finish the proof. □

The next theorem deals with case (IV).

Theorem 2.4. Let 𝒫 be a smooth projective plane of dimension n = 2^{k} = 2l. Put d_{IV} (1) = 2, d_{IV} (2) = 3, d_{IV} (3) = 7, and d_{IV} (4) = 14. If every minimal connected commutative closed normal subgroup N of Δ has some orbit x^{N} (x ∈ P ∪ℒ) which generates a subplane ℰ of 𝒫, then the dimension of the solvable group Δ is at most d_{IV} (k).

Proof: By M. Lüneburg, [17], Anhang B, Satz 3, the group Δ fixes some point or some line. Up to duality we may assume that Δ fixes some point p. Consider the derivation map D_{p} : Δ → Σ_{0}. Then the following assertions (a) and (b) hold:

(a) dim ker D_{p} = 0: If dim ker D_{p} > 0, then there is a minimal connected commutative closed normal subgroup M in ker D_{p} (namely a suitable commutator subgroup of (ker D_{p})^{1}, since ker D_{p} is a solvable group). By [2], (3.9), we have ker D_{p} = Δ_{[p,p]}. Hence the normal subgroup M cannot have an orbit (neither in P nor in ℒ) that generates a subplane of 𝒫. This contradicts the assumptions of the theorem, and thus we have dim ker D_{p} = 0. From (a) and Lemma 1.2 we get

dim Δ = dim D_{p}Δ.

(b) dim Δ_{[p]} = 0: Clearly, Δ_{[p]} is a normal subgroup of Δ. Since the connected component of the identity Δ_{[p]}^{1} is a characteristic subgroup of Δ_{[p]}, the connected group Δ_{[p]}^{1} is a normal subgroup of Δ, too. Assume that Δ_{[p]}^{1} ≠ 1l. According to part (a), the group Δ_{[p]}^{1} contains no elations, i.e. Δ_{[p]}^{1} consists of homologies only. Since Δ is solvable and because locally compact groups of homologies are isomorphic to some closed connected subgroup of ℍ^{×} (see [2], (3.12)), we conclude that Δ_{[p]}^{1} is commutative (in fact, it is a subgroup of ℂ^{×}). Thus, the group Δ has a minimal connected commutative closed normal subgroup M consisting entirely of homologies. As in part (a), we infer that M has no orbit that generates a subplane of 𝒫. This contradiction shows that dim Δ_{[p]} = 0. From dim Δ_{[p]} = 0 we conclude that D_{p}Δ has no nontrivial connected subgroup consisting of homologies with center 0. In the following we put Σ := Aut(𝒜_{p})^{1}. Case k = 1. We know that D_{p}Δ is a subgroup of GL_{2}ℝ. If rk D_{p}Δ = 1, this implies that dim D_{p}Δ ≤ 2. Hence, we may assume that rk D_{p}Δ = 0. Since D_{p}Δ is solvable, we conclude that D_{p}Δ is conjugate to some subgroup of the 3-dimensional group ∇ of all (upper, say) triangular matrices of GL_{2}ℝ. If dim D_{p}Δ = 3, this implies that D_{p}Δ is conjugate to ∇^{1} (recall that Δ is connected). In particular, the group D_{p}Δ contains the connected set Λ := {λ · id | λ > 0} which consists of homologies with center 0, a contradiction. This shows that dim D_{p}Δ ≤ 2. Case k = 2. According to [25], 73.4, the group D_{p}Δ fixes some point ∞ on L_{∞}. Thus the line T_{p}L := ∞∨0 is fixed by D_{p}Δ as well. We assume that dim D_{p}Δ ≥ 4. Then dim Σ ≥ 8 follows. By the clasification of four-dimensional translation planes, see [25], 73.22, the plane 𝒜_{p} is either isomorphic to the complex affine plane 𝒜_{2}ℂ or to the translation plane 𝒜_{0} described in [25], 73.10(b). The automorphism group of the affine plane 𝒜_{0} has dimension 8, whence we get D_{p}Δ = Σ, because D_{p}Δ is connected. In particular, this shows that D_{p}Δ contains a one-parameter subgroup of homologies with center 0 and we end up again with a contradiction. So let 𝒜_{p} ≅ 𝒜_{2}ℂ and denote by K ≅ 𝕋^{r} a maximal compact subgroup of D_{p}Δ. Then Σ ≅ GL_{2}ℂ. By Proposition 1.3 we have r := rk D_{p}Δ ≤ k = 2. If r = 2, then K ≅ 𝕋^{2}. Since the set of fixed lines of K on the spread 𝒮_{p} ≅ 𝕊_{2} is nonempty, there exists a second fixed line T_{p}K of K in 𝒮_{p}, see Floyd [6], (4.1) and [5], (5.2). Hence K is a compact subgroup of the group D of all complex diagonal matrices. This implies that K = {diag(e^{is}, e^{it}) | s, t ∈ ℝ}. Thus the group K contains a one-parameter subgroup of homologies with center O, which is again a contradiction. This implies that r ≤ 1. The group K acts nontrivially on the spread 𝒮_{p} or K = 1l, for otherwise K would consist of homologies with center 0. Thus, if K ≠ 1l, then there is some line T_{p}K ∈ 𝒮_{p} which is moved by K. If r = 1, we even get K_{T_{p}K} = 1l. Of course, this is also true in the case r = 0. Hence, if r ≤ 1, then the torus rank of Ψ = (D_{p}Δ)_{T_{p}K}^{1} is zero (i.e. Ψ has no non-trivial compact subgroup) and we have

dim Ψ = dim D_{p}Δ − dim(T_{p}K)^{D_{p}Δ} ≥ 4 − 2 = 2.

Since Ψ fixes two distinct lines of 𝒜_{p} through 0, this group is again a subgroup of D. Unfortunately, we do not know a priori whether or not Ψ is a closed subgroup of D. If it is closed in D, then Ψ is in fact contained in the two-dimensional subgroup D_{ℝ} of D consisting of real diagonal matrices. From dim Ψ ≥ 2 we infer that Ψ = D_{ℝ}^{1}. In particular, the group Ψ contains the one-parameter group Λ which consists of homologies and the proof is completed. It remains to show that Ψ is closed in D. Assume the contrary. Then there exists a closed non-compact one-parameter subgroup P = {ρ(t) | t ∈ ℝ} of Δ_{K} which is mapped by D_{p} onto a non-closed one-parameter subgroup D_{p}P of Ψ (see Hochschild [13], Chap. XVI, Theorem 2.4 and recall that D_{p} is a covering map since dim ker D_{p} = dim Δ_{[p,p]} = 0) which can be represented as the group {diag(e^{it}, e^{iαt}) | t ∈ ℝ} of diagonal matrices for some irrational real number α. Putting D_{ℂ} = {diag(e^{it}, e^{it}) | t ∈ ℝ}, we see that the group D_{p}P ∩ D_{ℂ} is a dense subgroup of the homology group D_{ℂ}. In particular, there is a sequence (t_{n})_{n≥1} of real numbers with t_{n} → ∞ as n → ∞ such that σ_{n} := D_{p}(ρ(t_{n})) ∈ P ∩ D_{ℂ} which converges to the identity. Of course, the collineations ρ(t_{n}) are homologies of 𝒫 with center p. Since the derivatives at the point p of these homologies converge to the identity mapping, we may choose compact neighborhoods V and W of p, where V does not intersect any axis of the homologies ρ(t_{n}), such that ⋃_{n} W^{σ_{n}} ⊆ V. Now we can apply a lemma of Löwen, [16], (3.8) which states that for any point q ∈ W and any accumulation point r of q^{σ_{n}} there is a collineation σ which is an accumulation point of the sequence (ρ(t_{n})) and which maps q to r. But the one-parameter group P is closed in Δ and ρ(t_{n}) ∈ P does not accumulate at all because of t_{n} → ∞. This contradiction shows that Ψ is closed and we eventually have finished the case k = 2.

Case k = 3. We assume that dim D_{p}Δ ≥ 8. Then dim Aut(𝒜_{p}) ≥ 16 follows. In case of equality we get D_{p}Δ = Σ := Aut(𝒜_{p})^{1} and thus Δ contains a one-parameter group of homologies with center p, a contradiction. Hence, we may assume that dim Σ ≥ 17. Thus, the tangent translation plane 𝒜_{p} is known, see the list in [25], 82.25 or [11]. The following case by case study is based on this list. If 𝒜_{p} is the classical affine plane over the quaternions, then dim Δ_{[p]} ≥ 1 by 1.6 which is a contradiction. If dim Σ = 18, then Σ contains a subgroup isomorphic to SU_{2}ℂ. Hence, any maximal solvable subgroup A of Σ has dimension at most 16 and contains the one-parameter subgroup of homologies with center 0 and we get a contradiction as before. If dim Σ = 17, then either the kernel of 𝒜_{p} is isomorphic to ℂ (cases (iii), (iv) and (vi) of 82.25) and we end up with dim Δ_{[p]} ≥ 1 again, or Σ contains a subgroup isomorphic to SO_{3}ℝ. In the latter case we conclude that dim A ≤ 15 for any maximal solvable subgroup A of Σ which is again a contradiction. Thus, dim D_{p}Δ ≤ 7 holds.

Case k = 4. As in the previous theorem we apply Lemma 2.2 in order to prove the claim. □

The remaining case (III) that may occur for solvable groups can be deduced from the three preceding theorems by means of induction if we recall that the kernel of the action of Δ on a Baer subplane is at most one-dimensional (zero-dimensional in the case dim 𝒫 = 4), see M. Lüneburg, [17], I, V.5.

Theorem 2.5. Let 𝒫 be a smooth projective plane of dimension 2^{k} ≥ 4. Assume that the solvable group Δ leaves some Baer subplane ℬ invariant. Put d_{III}(2) = 3, d_{III}(3) = 9, and d_{III}(4) = 17. Then the dimension of Δ is at most d_{III}(k).

By considering the automorphism groups of the four classical planes it can easily be seen that the upper bounds for dim Δ are sharp bounds in every case excluding the ”otherwise” case which does not occur in the classical planes. Summing up the theorems of this section we get the following table for a non-classical smooth projective plane.

dim 𝒫  flag  antiflag  Baer subplane  otherwise
2  3  3  —  2
4  8  6  3  3
8  min {16, 13 + t}  9  9  7
16  min {29, 22 + t}  14  17  14

In order to compare our results with the results in the topological case, we also list the bounds of M. Lüneburg [17], p. 36:

dim 𝒫  flag  antiflag  Baer subplane  otherwise
2  5  3  —  4
4  10  6  5  6
8  17  13  11  13
16  30  24  18  24

References

[1] Bödi, R.: Smooth stable planes. (to appear in Results Math.)

[2] Bödi, R.: Collineations of smooth stable planes. (submitted to Forum Math.)

[3] Bödi, R.: Stabilizers of collineation groups of smooth stable planes. (submitted to Indag. Math.)

[4] Buchanan, T.; Hähl, H.: On the kernel and the nuclei of 8–dimensional locally compact quasifields. Arch. Math. 29 (1977), 472–480

[5] Floyd, E.E.: On periodic maps and the Euler characteristic of associated spaces. Trans. Amer. Math. Soc. 72 (1952), 138–147

[6] Floyd, E.E.: Orbits of torus groups acting on manifolds. Ann. of Math. 65 (1957), 505–512

[7] Grundhöfer, T.; Salzmann, H.: Locally compact double loops and ternary fields. In: O. Chein, H.D. Pflugfelder, J.D.H. Smith (Eds.), Quasigroups and Loops: Theory and Applications, Berlin: Heldermann, 1990

[8] Grundhöfer, T.; Hähl, H.: Fibrations of spheres by great spheres over division algebras and their differentiability. J. Differential Geometry 31 (1990), 357–363

[9] Hähl, H.: Automorphismengruppen von lokalkompakten zusammenhängenden Quasikörpern und Translationsebenen. Geom. Dedicata 4 (1975), 305–321

[10] Hähl, H.: Homologies and elations in compact, connected projective planes. Topology Appl. 12 (1981), 49–63

[11] Hähl, H.: Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17dimensionaler Kollineationsgruppe. Geom. Dedicata 21 (1986), 299-340

[12] Halder, H.R.: Dimension der Bahn lokal kompakter Gruppen. Arch. Math. 22 (1971), 302–303

[13] Hochschild, G.: The structure of Lie groups. San Francisco, London, Amsterdam: Holden-Day, 1965

[14] Löwen, R.: Vierdimensionale stabile Ebenen. Geom. Dedicata 5 (1976), 239–294 [15] Löwen, R.: Topology and dimension of stable planes: On a conjecture of H. Freuden-

thal. J. Reine Angew. Math. 343 (1983), 108–122

[16] Löwen, R.: Central collineations and the parallel axiom in stable planes. Geom. Dedicata 10 (1981), 283–315

[17] Lüneburg, M.: Involutionen, auflösbare Gruppen und die Klassifikation topologischer Ebenen. Mitt. Math. Sem. Giessen 209 (1992), 1–113 [18] Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. of Math. 44

(1943), 454–470

[19] Montgomery, D., Zippin, L.: Topological transformation groups. New York: Interscience Publ., Inc., 1955

[20] Nagami, K.: Dimension-theoretical structure of locally compact groups. J. Math. Soc. Japan 14 (1962), 379–390

[21] Otte, J.: Differenzierbare Ebenen. Dissertation Kiel, 1992

[22] Otte, J.: Smooth Projective Translation Planes. Geom. Dedicata 58 (1995), 203–212 [23] Pickert, H.: Projektive Ebenen. Springer Verlag, 1975

[24] Salzmann, H.: Topological Planes. Advances in Math. 2 (1967), 1–60

[25] Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M.: Compact Projective Planes. De Gruyter, 1995

[26] Stroppel, M.: Achtdimensionale stabile Ebenen mit quasieinfacher Automorphismengruppe. Dissertation, Tübingen, 1990

[27] Stroppel, M.: Stable planes with large groups of automorphisms: the interplay of incidence, topology, and homogeneity. Habilitationsschrift, Darmstadt, 1993

Received: April 24, 1997