Journal of Lie Theory Volume 9 (1999) 481–486 C ◯1999 Heldermann Verlag

Large automorphism groups of 16-dimensional planes are Lie groups, II

Helmut Salzmann

Communicated by K. H. Hofmann

Abstract. Let P be a compact, 16-dimensional projective plane. If the group Σ of all continuous collineations of P is taken with the compact-open topology, then Σ is a locally compact group with a countable basis. The following theorem is proved: If the topological dimension dim Σ is at least 29, then Σ is a Lie group.

The automorphism group Σ of a projective plane P with compact, 16-dimen-sional point space P is a locally compact transformation group of P, and Σ has a countable basis [9, 44.3]. It is an open problem whether or not Σ is always a Lie group. If the topological dimension dim Σ is sufficiently large and if Σ is a Lie group, then the structure theory for Lie groups can be exploited to determine all possible planes. This has successfully been done in several cases, cp. [9, Chap. 8] and [8]. Therefore, the following criterion is useful:

Theorem. If dim Σ ≥ 29, then Σ is a Lie group.

In order to conclude that the connected component Σ^{1} of Σ is a Lie group, a weaker hypothesis suffices [7]:

If dim Σ ≥ 27, then Σ^{1} is a Lie group.

A theorem of Bödi [1], Proposition G in [7], and [9, 53.2] imply

(□) If Σ is not a Lie group, and if the subgroup Λ of Σ fixes a quadrangle, then dim Λ ≤ 11. Moreover, dim x^{Σ} = dim Σ/Σ_{x} < 16 for each point x.

The next result has been stated in [7, (a)] for connected subgroups of Σ, but the proof does not use connectedness:

Proposition. If Δ leaves some proper closed subplane invariant, then dim Δ ≤ 25 or Δ is a Lie group.

All large semi-simple groups on a 16-dimensional plane P are known [5], [6]:

If dim Δ > 28 and if Δ^{1} is semi-simple, then either P is a Hughes plane (including the classical Moufang plane), or Δ^{1} ≅ Spin_{9}(ℝ, r) with r ≤ 1.

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag

The full group of a Hughes plane is a Lie group [9, 86.12 and 53.2] or [9, 86.35]. The groups Spin_{9}(ℝ, r) contain the 28-dimensional compact group Spin_{8}ℝ (which fixes a triangle), and Σ is a Lie group by (□):

Corollary. If dim Δ > 28 and if Δ^{1} is semi-simple, then Σ is a Lie group.

The proof of the theorem uses the approximation theorem [9, 93.8] for locally compact groups: there is an open subgroup Δ of Σ and an arbitrarily small compact, 0-dimensional normal subgroup Θ ◁ Δ such that Δ/Θ is a Lie group. According to [9, 93.18], the connected component Δ^{1} = Σ^{1} and the group Θ centralize each other, and Δ_{a}^{1} acts trivially on the orbit a^{Θ} . A group Ξ is called straight if each point orbit x^{Ξ} is contained in a line, and a well-known theorem of Baer implies that either Ξ is planar (i.e. the fixed elements of Ξ form an 8-dimensional subplane F_{Ξ} ), or Ξ is contained in a group Σ_{[z]} of collineations with common center z, see [7, Th.B].

Assume now that dim Σ ≥ 29 and that Σ is not a Lie group, and choose Δ and Θ as above. Then Θ is not a Lie group, and the Proposition shows that Θ cannot be planar. By Baer’s Theorem, there remain two possibilities: either Θ is not straight and some orbit a^{Θ} contains a triangle, or Θ consists of axial collineations with a common center. Note that Δ^{1} is not semi-simple by the above Corollary.

(i) If a^{Θ} consists of more than 3 non-collinear points, then a^{Θ} generates a subplane, and (□) implies dim Δ_{a}^{1} ≤ 11, dim Δ ≤ 26. If a^{Θ} is just a triangle, however, and if the same is true for all orbits b^{Θ} with b near a, then a^{Θ} ∪b^{Θ} = C generates a subplane, Θ induces on C a finite group Θ/Λ, the kernel Λ is not a Lie group, Λ ≠ 1l, and F_{Λ} would be a Δ^{1}Λ-invariant proper closed subplane. This contradicts the Proposition. Hence Θ must be straight.

(ii) Because all arguments can be dualized, the elements of Θ also have a common axis W, and Θ is contained either in a group Σ_{[a,W]} of homologies (a ∉ W), or in a group Σ_{[v,W]} of elations with center v ∈ W. The case that Θ consists of homologies and that Δ is connected has been treated in [7]. A contradiction is obtained by studying the possible actions of the Lie group Δ/Θ on the axis W. The reasoning remains valid, if instead of the center Z of Δ the centralizer of Δ^{1} in Δ is used throughout. In the remaining case Θ ≤ Σ_{[v,W]} , the situation is different; it is the only one, in which the stronger hypothesis dim Σ ≥ 29 is needed. If Δ is connected, a theorem of Löwen [3] implies that Δ is a Lie group regardless of its dimension, cp. [4, (2.7)]. There seems to be no way, however, to extend Löwen’s proof to non-connected groups. In the general case, a proof can be based on a careful analysis of a point stabilizer.

(1) Suppose again that Θ ≤ Σ_{[v,W]} with v ∈ W. Choose any point a ∉ W, and consider the connected component Γ of Δ_{a}. Because Γ ∩ Θ = 1l, there is an embedding of Γ into the Lie group Δ/Θ. Hence Γ is itself a Lie group, and Γ has a minimal commutative, connected normal subgroup Ξ, or Γ is semisimple. As has been noted before, Γ fixes the (infinite) orbit a^{Θ} pointwise. The dimension formula [9, 96.10] and (□) imply 14 ≤ g = dim Γ ≤ 26. Moreover, Γ acts effectively on W, and there is at most one point u ∈ W \ {v} such that u^{Γ} = u, compare [7, Prop. G] (2) Let E = ⟨a^{Θ}, z^{Γ}⟩ denote the smallest closed subplane containing the orbits a^{Θ} and z^{Γ} . If z ∈ W \ {v} and z ≠ u, then z^{Γ} is a non-trivial connected set, and E has dimension d ∈ {2, 4, 8, 16}, see [9, 54.11]. Remember that z^{Θ} = z and that Γ and Θ commute. Consequently, E^{Θ} = E. As a group of elations, Θ acts effectively on E. Since each automorphism group of a plane of dimension d ≤ 4 is a Lie group [9, 32.21 and 71.2], it follows that E is a Baer subplane or the plane P itself, for short, E ≤^{•} P.

(3) Similarly, if Π is a one-parameter subgroup of Γ and if z^{Π} ≠ z, then ⟨a^{Θ}, z^{Π}⟩ ≤^{•} P. Let Ψ denote the connected component of the centralizer of Π in Γ. The Lie group Ψ_{z} acts trivially on ⟨a^{Θ}, z^{Π}⟩, and Ψ_{z}^{1} is isomorphic to a subgroup of SU_{2}ℂ by [9, 83.22]. In particular, dim Ψ_{z} ≤ 3, dim Ψ ≤ 11. Note that CsΠ = Csϱ = Γ_{ϱ} for any ϱ ∈ Π \ {1l}. The dimension formula [9, 96.10] gives g − 8 ≤ dim Γ_{z} ≤ dim ϱ^{Γ_{z}} + 3.

(4) Because a compact, commutative normal subgroup of Γ is contained in the center, it follows from (1) and (3) that either Γ is semi-simple, or Γ has a minimal normal subgroup Ξ ≅ ℝ^{t} with t ≥ g − 11, compare [9, 94.26]. The semi-simple case will be discussed later.

(5) Assume that ℝ^{t} ≅ Ξ◁ Γ, and let z^{Γ} ≠ z ∈ W \{v}. If z^{Ξ} = z, then Ξ induces the identity on E ≤^{•} P, and Ξ would be compact by [9, 83.6]. Consequently, z^{Ξ} ≠ z, and ⟨a^{Θ}, z^{Ξ}⟩ ≤^{•} P by the arguments of (2). Since Ξ_{z} fixes each point of ⟨a^{Θ}, z^{Ξ}⟩, it follows that Ξ_{z} is compact, and then Ξ_{z} = 1l. Therefore, Ξ acts freely on W \ {u, v} or on W \ {v}, and t ≤ 8, g ≤ 19. In particular, dim a^{Δ} ≥ 10, and the line av is not fixed by Δ.

(6) If g = 19, and if 1l ≠ ϱ ∈ Ξ, then (3) implies dim Γ_{z} = 11, dim ϱ^{Γ_{z}} = 8, and ϱ^{Γ_{z}} is open in Ξ by [9, 92.14 or 96.11(a)]. Hence Γ_{z} is transitive on Ξ \ {1l}, and a maximal compact, connected subgroup is transitive on the 7-sphere of the rays in Ξ ≅ ℝ^{8} , see [9, 96.19]. With [9, 96.20–22] it follows that Γ′_{z} ≅ U_{2}ℍ. The central involution σ ∈ Γ_{z} inverts each element of Ξ, and z is an isolated fixed point of σ on W. Therefore, σ is a reflection with center z and axis av. This contradicts the following Lemma on involutions, which will be needed repeatedly:

(∗) Let α, β, and αβ be pairwise commuting involutions in Γ. If Θ is not a Lie group, then exactly one of the 3 involutions is a reflection, and the torus rank rk Γ ≤ 2. Each reflection in Γ has axis av and some center z ∈ W. Moreover, Γ has no subgroup Φ ≅ SO_{3}ℝ, and dim z^{Γ} ≤ 6. Proof. Any involution is either a reflection, or it is planar [9, 55.29]. If all 3 involutions α, β, and αβ are planar, then the common fixed elements of α and β form a 4-dimensional subplane F, see [9, 55.39(a)]. By definition, Γ is connected, Γ and Θ centralize each other, and F^{Θ} = F. Because Θ consists of elations, Θ acts effectively on F, and Θ would be a Lie group by [9, 71.2]. Hence we may assume that α is a reflection. Because Γ fixes the orbit a^{Θ} pointwise, each reflection in Γ has axis av, its center z lies on the fixed line W. Since the center of one of two commuting reflections is on the axis of the other [9, 55.35], the involutions β and αβ are planar. If SO_{3}ℝ ≅ Φ ≤ Γ, and if α and β are chosen in Φ, then α and β are conjugate in Φ and therefore would be of the same kind, a contradiction. If dim z^{Γ} = k > 0, then α^{Γ}α is a k -dimensional set in the connected component E of the elation group Δ_{[v,av]} , compare [9,61.19(b)]. The last statement in (5) implies that E is commutative, in fact, E ≅ ℝ^{k}. The connected group Γ induces linear maps of positive determinant on E. In particular, det α = 1. On the other hand, the reflection α inverts each element in E, and α|_{E} = −1l. Consequently, k is even. If k = 8, then Δ_{[v,v]} is transitive, and Θ would be contained in the Lie group Δ_{[v,W]} ≅ ℝ^{8} . This completes the proof of Lemma (∗).

(7) Now let 14 ≤ g ≤ 18, and put Ω = Γ_{z}^{1} . Then dim Ω ≥ 6, and for each ϱ ∈ Ξ \ {1l} the last assertion in (3) gives dim ϱ^{Ω} ≥ 3. Hence any minimal Ωinvariant subspace Υ ≤ Ξ has dimension s ≥ 3. If s = 3, then Ω is transitive on Υ \ {1l}, and Ω_{ϱ} ≅ SU_{2}ℂ by (3). Because Ω_{ϱ} fixes a subspace of Υ, the representation of Ω_{ϱ} on Υ is trivial. Consequently, Ω/Ω_{ϱ} would act sharply transitive on Υ \ {1l} ≅ ℝ^{3} \ {0}, but such a group does not exist.

(8) Similarly, the case s = 4 leads to a contradiction: because SU_{2}ℂ has no 2-dimensional subgroup, one has again Ω_{ϱ} ≅ SU_{2}ℂ for each ϱ ∈ Υ \ {1l}. Being compact, Ω_{ϱ} acts on Υ as an orthogonal group, in fact as a subgroup of SO_{3}ℝ. Hence the central involution ω of Ω_{ϱ} is planar, the Baer subplane of its fixed elements is F_{ω} = ⟨a^{Θ}, z^{Υ}⟩. Either Ω_{ϱ} acts trivially on F_{ω}, or Ω_{ϱ} induces on F_{ω} a group Φ = Ω_{ϱ}/⟨ω⟩ ≅ SO_{3}ℝ. In the latter case, Φ fixes a quadrangle in F_{ω} by its very definition. It follows that the fixed elements of Φ in F_{ω} form a 2-dimensional subplane (use [9, 96.34]). Acting faithfully on this subplane, Θ would be a Lie group by [9, 32.21]. Therefore, Ω_{ϱ} is the kernel of the irreducible action of Ω on Υ, and (Ω/Ω_{ϱ})′ is a non-trivial semi-simple linear group. Consequently, Ω′ contains a 2-torus. Lemma (∗) implies dim z^{Γ} ≤ 6, but then 14 ≤ g ≤ dim z^{Γ} + dim Ω ≤ 6 + s + 3 = 13. This contradiction shows that s > 4.

(9) By the last assertion, Ω = Γ_{z}^{1} acts faithfully and irreducibly on Υ ≅ ℝ^{s}, and the semi-simple commutator subgroup satisfies dim Ω′ > 3, hence dim Ω′ ≥ 6, see [9, 95.6]. If s ∈ {5, 7}, then Γ′_{z} is almost simple and irreducible on Υ by Clifford’s Lemma [9, 95.5]. Inspection of a list of irreducible representations [9, 95.10] shows that either s = 5 and dim Γ′_{z} ≥ 10, or s = 7 and dim Γ′_{z} ≥ 14, but dim Γ_{z} ≤ s + 3. Hence s ∈ {6, 8}.

(10) Suppose that s = 6 = dim Ω′. Lemma (∗) implies rk Ω = 1, or rk Ω = 2 and dim z^{Γ} ≤ 6. In the second case, dim Ω = 8, and the center of Ω is isomorphic to ℂ^{×}. Consequently, rk Ω′ = 1 and Ω′ is almost simple and locally isomorphic to SL_{2}ℂ. From [9, 95.6(b) and 95.10] it follows that Ω′ acts irreducibly on Υ and Ω′ ≅ SO_{3}ℂ > SO_{3}ℝ. This contradicts (∗).

(11) If s = 6 and dim Ω′ = 8, then Ω′ is isomorphic to a group SU_{3}(ℂ, r) or to SL_{3}ℝ. None of these groups contains a central involution. Consequently, each involution in Ω′ has a positive eigenspace in Υ and hence is planar. Moreover, there are 3 pairwise commuting involutions in Ω′. This is excluded by (∗).

(12) The case s = 6 and dim Ω′ = 9 leads to a contradiction as follows: a 9-dimensional semi-simple group is not almost simple and has at least one 3dimensional factor. On the other hand, the arguments of (6) show that Ω′ acts transitively on Υ \ {1l} and hence on the 5-sphere consisting of the rays in Υ ≅ ℝ^{6} . Therefore, Ω′ contains an 8-dimensional almost simple factor SU_{3}ℂ.

(13) From (7–12), it follows that Υ = Ξ ≅ ℝ^{8} . If z ∈ W \ {v}, z ≠ u, then z^{Ξ} ≈ ℝ^{8} by step (5), and z^{Ξ} is open in W by [9, 53.1(a)]. Hence W is a manifold, and W ≈ 𝕊_{8} according to [9, 52.3]. Since W \ {u, v} ≉ ℝ^{8} , the group Ξ is sharply transitive on W \ {v}. Remember that Γ_{z} acts effectively on Ξ.

(14) Combination of (13) and (∗) shows that the group Ω = Γ_{z} does not contain any reflection. The semi-simple commutator subgroup Ω′ has dimension at least 6. Because of (∗), its torus rank is 1, and Ω′ is even almost simple. The only groups satisfying these conditions and having a faithful linear representation are SL_{2}ℂ, SO_{3}ℂ, and SL_{3}ℝ, see [9, 95.10]. In the first case, the central involution would be a reflection. The latter two groups have a subgroup SO_{3}ℝ and hence are excluded by (∗). Together, steps (4–14) imply that Γ is semi-simple.

(15) If Γ has two or more factors, choose an almost simple factor B of maximal dimension and let A denote the product of the other factors, so that A and B commute elementwise. Consider z ∈ W with z^{Γ} ≠ z and ⟨a^{Θ}, z^{Γ}⟩ = E ≤^{•} P as in (2). Assume first that z^{A} = z. Then A acts trivially on E and E is a Baer subplane, moreover, A ≅ SU_{2}ℂ by [9, 83.22]. Therefore, dim B ≥ 11. Since B is almost simple, dim B ≥ 14 and B acts almost effectively (i.e. with discrete kernel) on E. But B fixes a^{Θ} , and the stiffness theorem [9, 83.17] gives dim B ≤ 7 + 4, a contradiction. Similarly, z^{B} = z implies dim B = 3, and A is a product of 3-dimensional groups by the maximality of B. Hence dim A ≥ 12. The kernel K of the action of Γ on E contains B, and dim K = 3 by [9, 83.22]. Consequently, A acts almost effectively on E. Again, the stiffness theorem shows dim A ≤ 11. Thus, ⟨a^{Θ}, z^{A}⟩ = A ≤^{•} P and ⟨a^{Θ}, z^{B}⟩ = B ≤^{•} P.

(16) As in step (3), the last part of (15) implies dim A_{z} ≤ 3 and dim A ≤ 11. If dim B ≤ 6, then dim A ≡ 0 mod 3 and dim A = 9. Therefore, dim z^{A} ≥ 6 and A = P. Consequently, B_{z} = 1l, dim B = 6, and B = P. Now A_{z} = 1l and dim A ≤ 8, a contradiction. Since also dim B ≤ 11 and B is almost simple, it follows that dim B ∈ {8 , 10} and dim z^{B} > 4. Hence B = P and again A_{z} = 1l. Because dim Γ ≥ 14, the semi-simple group A has dimension at least 6, and A = P, so that B_{z} = 1l and dim B = 8.

(17) By [9, 53.1(a)], the orbit z^{B} is open in W whenever z^{Γ} ≠ z, and this is true for each point z ∈ W \ {v} with at most one exception u, see step (1). Hence B is sharply transitive on W \ {v} ≈ ℝ^{8} or on W \ {u, v} ≈ e^{ℝ} × 𝕊_{7} . In both cases, the homotopy group π_{3}B vanishes, but every almost simple Lie group X satisfies π_{3}X ≅ ℤ, see [2] or [9, 94.36]. Therefore, Γ is almost simple.

(18) If the center Z of Γ is not trivial, and if z^{Z} ≠ z ∈ W, then Γ_{z} fixes each point of ⟨a^{Θ}, z^{Z}⟩, and (□) implies dim Γ_{z} ≤ 11, dim Γ < 20. Therefore, Γ is of type G_{2} , or Γ is locally isomorphic to one of the groups SU_{4}(ℂ, r), SL_{2}ℍ, SL_{4}ℝ, SL_{3}ℂ, or dim Γ ≥ 20 and Γ is even simple in the strict sence, cp. [9, 94.21]. In any case, Γ has a compact subgroup Φ which is locally isomorphic to SU_{3}ℂ or to (SU_{2}ℂ)^{2} . Note that SO_{3}ℝ < SU_{3}ℂ. Hence Φ contains a subgroup SO_{3}ℝ or Φ = A × B with A ≅ B ≅ SU_{2}ℂ. The first possibility is excluded by Lemma (∗).

(19) Finally, consider the alternative Γ > Φ = A × B of the last step, and let α ∈ A and β ∈ B be the central involutions of the two factors. Assume that β is not a reflection (∗). Then the fixed elements of β form a ΦΘ-invariant Baer subplane B, and a^{Θ} ⊆ K = av ∩ B. Lemma (∗) implies that α acts on B as a reflection with axis K and some center z ∈ W. Because a compact group of (z, K)-homologies of B has dimension at most 3, the group Φ acts non-trivially on K. Since Φ fixes each point of a^{Θ} , it follows from Richardson’s theorem [9, 96.34] that Φ induces on K a group SO_{3}ℝ, and that the fixed points of Φ on K form a circle S. The group Θ acts effectively on S and hence would be a Lie group. This contradiction completes the proof of the theorem.

References

[1] Bödi, R., On the dimensions of automorphism groups of eight-dimen-sional ternary fields I, J. Geom. 52 (1995), 30–40.

[2] Bott, R., An application of the Morse Theory to the topology of Liegroups, Bull. Soc. Math. France 84 (1956), 251–281.

[3] Löwen, R., Locally compact connected groups acting on euclidean space with Lie isotropy groups are Lie, Geom. Dedicata 5 (1976), 171–174.

[4] Löwen, R., and H. Salzmann, Collineation groups of compact connected projective planes, Arch. Math. 38 (1982), 368–373.

[5] Priwitzer, B., Large semisimple groups on 16-dimensional compact projective planes are almost simple, Arch. Math. 68 (1997), 430–440.

[6] —, Large almost simple groups acting on 16-dimensional compact projective planes, Monatsh. Math., to appear.

[7] Priwitzer, B., and H. Salzmann, Large automorphism groups of 16dimensional planes are Lie groups, J. Lie Theory 8 (1998), 83–93.

[8] Salzmann, H., Characterization of 16-dimensional Hughes planes, Arch. d. Math., 71 (1998), 249–256.

[9] Salzmann, H., D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, and M. Stroppel, “Compact projective planes,” W. de Gruyter, Berlin-New York, 1995.

Mathematisches Institut Auf der Morgenstelle 10 D – 72076 Tübingen e-mail: helmut.salzmann@uni-tuebingen.de

Received July 10, 1998 and in final form October 12, 1998