Large automorphism groups of 16-dimensional planes are Lie groups

It is a major problem in topological geometry to describe all compact projective planes P with an automorphism group Σ of sufficiently large topological dimension. This is greatly facilitated if the group is known to be a Lie group. Slightly improving a result from the first author’s dissertation, we show for a 16-dimensional plane P that the connected component of Σ is a Lie group if its dimension is at least 27. Compact connected projective planes P of finite topological dimension exist only in dimensions d = 2`|16, see [1], 54.11. In the compact-open topology, the automorphism group Σ of such a plane P is locally compact and has a countable basis [1], 44.3, its topological dimension dim Σ is a suitable measure for the homogeneity of P . The so-called critical dimension c` is defined as the largest number k such that there exist 2` -dimensional planes with dim Σ = k other than the classical Moufang plane over R , C , H , or O respectively, compare [1], § 65. Analogously, there is a critical dimension c̃` for smooth planes, and c̃` ≤ c` − 2 by recent work of Bödi [3]. The classification program requires to determine all planes P admitting a connected subgroup ∆ of Σ with dim ∆ sufficiently close to c` ; most results that have been obtained so far fall into the range 5` − 3 ≤ dim ∆ ≤ c` . Additional assumptions on the structure of ∆ or on its geometric action must be made for smaller values of dim ∆ . The cases ` ≤ 4 are understood fairly well. For ` = 8, however, results are still less complete, and we shall concentrate on 16dimensional planes from now on. It is known that c8 = 40, and all planes with dim Σ = 40 can be coordinatized by a so-called mutation of the octonion algebra O , see [1], 87.7. All translation planes with dim Σ ≥ 38 have been described explicitly by their quasi-fields [1], 82.28. If P is a proper translation plane, then Σ is an extension of the translation group T ∼= R by a linear group, in particular, Σ is then a Lie group. In her dissertation, the first author proved the following result under the hypothesis dim Σ ≥ 28. With only minor modifications, her proof yields ISSN 0949–5932 / $2.50 C © Heldermann Verlag 84 Priwitzer, Salzmann Theorem L. If dim Σ ≥ 27 , then the connected component Σ of Σ is a Lie group. This covers all known examples and all cases in which a classification might be hoped for. A weaker version of Theorem L is given in [1], 87.1, for 8-dimensional planes see also Priwitzer [5]. Here we shall present a proof of Theorem L. The whole structure theory of real Lie groups then becomes available for the classification of sufficiently homogeneous 16-dimensional planes. How such a classification can be achieved has been explained in [1], § 87, second part. Two of the results mentioned there have been improved considerably in the meantime: Theorem S. Let ∆ be a semi-simple group of automorphisms of the 16dimensional plane P . If dim ∆ > 28 , then P is the classical Moufang plane, or ∆ ∼= Spin9(R, r) and r ≤ 1 , or ∆ ∼= SL3H and P is a Hughes plane as described in [1], § 86 . The proof can be found in Priwitzer [6, 7]. Theorem T. Assume that ∆ has a normal torus subgroup Θ ∼= T . If dim ∆ > 30 , then Θ fixes a Baer subplane, ∆′ ∼= SL3H , and P is a Hughes plane. To prove Theorem L, we will use the approximation theorem as stated in [1], 93.8. The proof distinguishes between semi-simple groups and groups having a minimal connected, commutative normal subgroup Ξ , compare [1], 94.26. A result of Bödi [2] plays an essential role: Theorem Q. If the connected group Λ fixes a quadrangle, then Λ is isomorphic to the compact Lie group G2 , or dim Λ ≤ 11 . Moreover, dim Λ ≤ 8 if the fixed points of Λ form a 4-dimensional subplane. The last assertion follows from Salzmann [10], § 1, Corollary. In translation planes, the stabilizer Λ of a quadrangle is compact. Presumably, the same is true for compact, connected planes in general, but for 25 years all efforts have failed to prove compactness of Λ without additional assumptions. This causes some of the difficulties in the following proofs. Consider any connected subgroup ∆ of Σ . If the center Z of ∆ is contained in a group of translations with common axis (or with common center), then ∆ is a Lie group by Löwen – Salzmann [4] without any further assumption. Assume now that ∆ is not a Lie group. By the approximation theorem, there is a compact, 0-dimensional central subgroup Θ such that ∆/Θ is a Lie group. In particular, Θ ≤ Z is infinite. The elements of Z can act on the plane in different ways. This leads to several distinct cases. We say that the collineation η is straight if each orbit x〈η〉 is contained in a line, and η is called planar if the fixed elements of η form a proper subplane. By a theorem of Baer [1], 23. 15 and 16, a straight collineation is either planar or axial. Hence Theorem L is an immediate consequence of propositions (a—d) which will be proved in this paper. Priwitzer, Salzmann 85 (a) If ∆ leaves some proper closed subplane F invariant (in particular, if Z contains a planar element), or if ∆ is semi-simple, then dim ∆ < 26 . (b) If ζ ∈ Z is not straight, or if Z contains axial collineations with different centers, then dim ∆ ≤ 26 . (c) If dim ∆ > 26 , then Z is contained in a group ∆[a,W ] of homologies. Moreover, a minimal connected, commutative normal subgroup Ξ of ∆ is also contained in ∆[a,W ] . (d) If ΞZ ≤ ∆[a,W ] as in (c), then dim ∆ ≤ 26 , i.e. case (c) does not occur. The following criteria will be used repeatedly: Theorem O. If Σ has an open orbit in the point space, or if the stabilizer ΣL of some line L acts transitively on L, then Σ is a Lie group. (An orbit having the same dimension as the point space P is open in P .) For proofs see [1], 53.2 and 62.11. The addendum is a consequence of [1], 51.12 and 96.11(a). From Szenthe’s Theorem [1], 96.14 and again [1], 51.12 and 96.11(a) we infer Lemma O. If the stabilizer ∆L of a line L has an orbit X ⊆ L with dimX = dimL , then X is open in L , and the induced group ∆L|X ∼= ∆L/∆[X] is a Lie group. The next result holds without restriction on the dimension of the group: Theorem P. The full automorphism group of any 2or 4-dimensional compact plane is a Lie group of dimension at most 8 or 16 respectively. Proofs are given in [1], 32.21 and 71.2. In conjunction with Theorem Q we need Proposition G. If Σ contains a subgroup Γ ∼= G2 , and if Γ fixes some element of the plane, then Σ is a Lie group. Proof. Assume that Σ is not a Lie group and that Γ fixes the line W . Being simple, Γ acts faithfully on W by [1], 61.26. There are commuting involutions α and β in Γ , and all involutions in Γ are conjugate, see [1], 11.31. Each involution is either a reflection or a Baer involution [1], 55.29, and conjugate involutions are of the same kind. In the case of reflections, one of the involutions α, β , and αβ would have axis W by [1], 55.35, and Γ would not be effective on W . Hence all involutions are planar [1], 55.29. Because of [1], 55.39, the fixed subplanes Fα and Fβ intersect in a 4-dimensional plane F . By [1], 55.6, Note, the lines are 8-spheres, and repeated application of [1], 96.35 shows that the fixed elements of Γ form a 2-dimensional subplane E < F . Moreover, each point z ∈W \E has an orbit z ≈ S6 . By the approximation theorem [1], 93.8, some open subgroup of Σ contains a compact central subgroup Θ which is not a Lie group. According 86 Priwitzer, Salzmann to Theorem P, the group Θ induces a Lie group on F , and the kernel K = Θ[F ] is infinite. Now choose z ∈ W such that z belongs to F but not to E . Then z = z , and K fixes each point of z ≈ S6 (note that Γ ◦Θ = 1l). Since F and z together generate the whole plane, we get K = 1l. This contradiction proves the proposition. Finally, we mention a result of M. Lüneburg [1], 55.40 which excludes many semi-simple groups as possible subgroups of ∆ : Lemma R. The group SO5R is never contained in Σ . A group Λ of automorphisms is called straight if each point orbit x is contained in some line. Baer’s theorem mentioned above is true in general for groups which are straight and dually straight. In compact planes of finite positive dimension 2` it holds in the following form: Theorem B. If Λ is straight, then Λ is contained in a group Σ[z] of central collineations with common center z , or the fixed elements of Λ form a Baer subplane FΛ . Proof. If all fixed points of Λ with at most one exception lie on one line, then the unique fixed line through any other point must pass through the same point z . If, on the other hand, there is a quadrangle of fixed points and Λ 6= 1l, then FΛ = (F,F) is a closed proper subplane. Suppose that FΛ is not a Baer subplane. By definition, this means that some line H does not meet the (Λ -invariant ) fixed point set F . For each x ∈ H the line Lx containing x is the unique fixed line through x . Choose p ∈ H and λ ∈ Λ with p 6= p . Then pp = Lp ∈ F and Lp 6= H 6= H (since H∩F = Ø and H / ∈ F). There is a compact neighbourhood V of p in H such that V ∩V λ = Ø. The map (x 7→ xx) : V → F is continuous and injective. Hence dim F = ` . This condition, however, characterizes Baer subplanes, see [1], 55.5. In the following, ∆ will always denote a connected locally compact group of automorphisms of a 16-dimensional compact projective plane P = (P,L). We assume again that Θ is a compact, 0-dimensional subgroup in the center Z of ∆ such that ∆/Θ is a Lie group but Θ is not. Groups of dimension ≥ 35 are known to be Lie groups [1], 87.1. Hence only the cases 25 < h = dim ∆ < 35 have to be considered. Proof of (a) (1) Assume that dim ∆ ≥ 26 and that F is any ∆ -invariant closed proper subplane. ∆ induces on F a group ∆∗ = ∆/Φ with kernel Φ . If dimF ≤ 4, then Theorems P and Q imply dim ∆ ≤ 24. Hence dimF = 8 and F is a Baer subplane. Moreover, the kernel Φ is compact and satisfies dim Φ < 8, see [1], 83.6. Consequently, dim ∆∗ ≥ 19, and then F is isomorphic to the classical quaternion plane P2H , cf. Salzmann [11] or [1], 84.28. In particular, ∆∗ is a Lie group, and we may assume Θ ≤ Φ . A semi-simple group ∆∗ in the given dimension range is, in fact, one of the simple motion groups PU3(H, r). This is proved in Salzmann [9], for almost simple groups cp. also [1], 84.19. Priwitzer, Salzmann 87 In all other cases, it has been shown in Salzmann [8] (4.8) that ∆ fixes some element of F , say a line W . The lines of P are homeomorphic to S8 because the point set of F is a manifold [1], 41.11(b) and 52.3. Any k -dimensional orbit in a k -dimensional manifold M is open in M , see [1], 92.14 or 96.11. Since ∆ is not a Lie group, Theorem O implies dim p < 16 for each point p . Moreover, we conclude from Lemma O that the stabilizer of a line of F has only orbits of dimension at most 7 on this line. The points and lines of F will be called “ inner ” elements, the others “ outer ” ones. There are outer points p and q not on the same inner line such that dim ∆/∆p,q ≤ dim p + dim q ≤ 15 + 7. (If ∆ fixes the inner line W , choose q ∈W ; if ∆∗ is a motion group corresponding to the polarity π of F ∼= P2H , and if p is on the inner line L = a , choose q on the line ap .) Hence the connected component Λ of ∆p,q satisfies dim Λ > 3. Because the infinite group Θ acts freely on the set of outer points, Λ ∩Θ = Θp = 1l, and Λ is a Lie group. The orbits p and q consist of fixed points of Λ , and all fixed elements of Λ form a proper subplane E . Since each outer line meets F in a unique inner point, E ∩ F is infinite. Any collineation group of P2H with 3 distinct fixed points on a line fixes even a point set homeomorphic to a circle on that line [1], 13.6 and 11.29. Therefore, dim E ∈ {2, 4, 8} . In the first two cases, Θ would be a Lie group by Theorem P. In the last case it follows from [1], 83.6 and 55.32(ii) that Λ is a compact Lie group of torus rank 1, and dim Λ ≤ 3. Thus, dim ∆ > 25 has led to a contradiction. (2) If ∆ is even almost simple, i.e. if ∆∗ = ∆/Z is simple, then ∆ is a projective limit of covering groups of ∆∗ , see Stroppel [12] Th. 8.3. In particular, the fundamental group π1∆ ∗ must be infinite. In the range 25 < h < 35 the last condition is satisfied only by ∆∗ ∼= PSO8(R, 2). Let Φ be a maximal compact subgroup of ∆ . The commutator subgroup Φ′ covers PSO6R . Lemma R implies Φ′ ∼= Spin6R ∼= SU4C . In SU4C there are 6 pairwise commuting diagonal involutions conjugate to α = diag (1, 1,−1− 1). Let β be one of these conjugates. From [1], 55. 34b and 39 together with [1], 55.29 it follows that the common fixed elements of α and β form a 4-dimensional subplane F . By Theorem P, the kernel K of the action of Θ on F is infinite. The subplane Q < P consisting of all fixed elements of K is ∆ -invariant (because Θ ≤ Z). On the other hand, it has been proved in [1], 84.9 that Φ′ cannot act on any proper subplane of P . This contradiction shows that a semi-simple group ∆ has at least two almost simple factors, cp. [1], 94.25. (3) Consider an almost simple factor A of ∆ of minimal dimension such that A is not a Lie group, and denote the product of all other factors by B . We will find successively smaller bounds for dim B . Write A∗ for the simple image of A in ∆∗ = ∆/Z . Let Φ be a maximal compact subgroup of A . The Mal’cevIwasawa theorem [1], 93.10 shows that A is homeomorphic to Φ×Rk , and Φ is not a Lie group. By Weyl’s theorem [1], 94.29, a compact semi-simple Lie group has only finitely many coverings. Hence Φ∗ cannot be semi-simple and has a central torus [1], 94.31(c). In fact, this central torus is one-dimensional as can be seen by inspection of the list of simple Lie groups [1], 94.33. Consequently, the connected component Υ of Z(Φ) is a 1-dimensional solenoid. In particular, A 6= Φ and A is not compact. In the next steps we will apply Theorem B to Υ and to Z . 88 Priwitzer, Salzmann (4) Assume first that Υ is straight, and let 1l 6= ζ ∈ Υ ∩ Z . If FΥ is a Baer subplane, then Fζ = FΥ would be a ∆ -invariant proper subplane in contradiction to (1). If Υ ≤ ∆[z] , then the center z of ζ is ∆ -invariant. In particular, z = z . Because A is almost simple and Υ is contained in the normal subgroup A[z] , we get A ≤ ∆[z] . Homologies and elations with center z or homologies with different axes and the same center do not commute. Hence Υ consists of elations only or of homologies with the same axis. If Υ is an elation group, so is A , and all elements in A have the same axis, because A is not commutative, cp. [1], 23.13. If Υ ≤ ∆[z,L] , then L is the axis of ζ , and L = L . Consequently, A[z,L] is a normal subgroup of A , and A ≤ ∆[z,L] . For z ∈ L the connected group A would be a Lie group [1], 61.5, and in the case z / ∈ L it would follow from [1], 61.2 that A is compact. This contradicts the last statement in (3). (5) Therefore, Υ is not straight, and there is some point c such that c generates a connected subplane. We shall write 〈cΥ〉 = F for the smallest closed subplane containing c . If dimF ≤ 4, then Υ induces a Lie group on F by Theorem P, and there is an element ζ ∈ Z such that F ≤ Fζ < P , but this contradicts (1). Thus, F is a Baer subplane or F = P . Since BΦ and Υ commute elementwise, (BΦ)c induces the identity on F , and dim(BΦ)c ≤ 7 by [1], 83.6. From Theorem O it follows that dim c ≤ 15. If dim c > 8, then 〈c∆〉 = P and Zc = 1l. Hence, (BΦ)c is a Lie group and we have even dim(BΦ)c ≤ 3 as at the end of (1). In any case, the dimension formula [1], 96.10 gives dim B + dim Φ ≤ 18 and dim A ≥ 8. Now the classification of simple Lie groups [1], 94.33 shows that dim Φ ≥ 4, and dim B ≤ 14. Consequently, dim A ≥ 12. The remarks in (3) and again the classification [1], 94.33 imply dim A ∈ {15, 21, 24} , and then dim Φ ≥ 7. We conclude that dim B ≤ 11, and B is a Lie group by the minimality assumption on dim A . (6) Suppose that Z is straight. FZ cannot be a Baer subplane by (1). Hence Z ≤ ∆[a] for some center a . If each element of Z is an elation, ∆ would be a Lie group by the dual of (2.7) in Löwen – Salzmann [4]. Therefore, the center Z is contained in a group ∆[a,W ] of homologies (note that homologies in ∆[a] with different axes do not commute). We can now show that ∆ has torus rank rk ∆ < 4. Else, it would follow from [1], 55.35 and 39 (a) that there are Baer involutions α and β in ∆ such that Fα,β is a 4-dimensional subplane. As a group of homologies, Z would act faithfully on Fα,β , but this contradicts Theorem P. At the end of step (5) we have seen that B is a Lie group of dimension at most 11. This implies dim A ≥ 15 and then rk A ≥ 2, see [1], 94. 32(e) or 33. If dim B = 11, then B is a product ΨΩ of two almost simple Lie groups such that dim Ψ = 8 and dim Ω = 3. It follows that rk Ψ = 1 and rk Ω = 0. Hence Ω is the universal covering group of SL2R , and Ω is not compact [1], 94.37. Since any almost simple subgroup of ∆[a,W ] is compact by [1], 61.2, the group Ω acts non-trivially on W , and there is a point x such that 〈xΩZ〉 = B is a connected subplane of P . Because Z consists of homologies, Z acts faithfully on B , and Theorem P shows that dimB ≥ 8, i.e. B is a Baer subplane, or B = P . The stabilizer Λ = (AΨ)x fixes B pointwise, moreover, Λ ∩ Z = 1l, and Λ is a Lie group. From [1], 83.6 and 55.32(ii) we conclude again that Λ is compact, Priwitzer, Salzmann 89 that rk Λ ≤ 1, and hence dim Λ ≤ 3. With dim Ψ = 8 we get dim A < 11, a contradiction. The only remaining possibility dim B < 11 and dim A ≥ 21 can be excluded by similar arguments: If B acts non-trivially on W , then B = 〈xBZ〉 is a subplane of dimension at least 8, and dim Ax ≤ 3, dim A < 20. If B ≤ ∆[a,W ] , however, then B is compact by [1], 61.2. At the end of (5) it has been stated that B is a Lie group, and we know also that rk B ≤ 1. Consequently, dim B = 3, dim A = 24, and rk A = 3, but we have proved above that rk ∆ < 4. (7) Finally, we consider the case that Z is not straight. There is a point c such that the orbit c is not contained in a line. In particular, 〈c∆〉 is a ∆ invariant subplane, and 〈c∆〉 = P by step (1). Hence Zc = 1l, and 〈cZ〉 is a non-degenerate subplane. By Theorems Q and G, we have dim ∆c ≤ 11, and we conclude from Theorem O that dim c < 16. The dimension formula gives dim ∆ = 26. If dim A > 15, then dim A ∈ {21, 24} and dim B ∈ {5, 2} , and B would not be semi-simple. Consequently, dim B = 11, and we have again that B is a product of two almost simple factors Ψ and Ω with dim Ω = 3. Let C be the set of all points x such that x is not contained in any line. Then C is an open neighborhood of c , and Ω|C 6= 1l. We may assume that c 6= c . Consider the subplane B = 〈cΩZ〉 . Because Zc = 1l, it follows as in step (6) that dimB ≥ 8, and then dim(AΨ) < 20. This contradiction completes the proof of (a). Proof of (b) By Theorem B and (a), each assumption implies that Z is not straight. As in step (7) above, some orbit c contains a quadrangle, and from Theorems Q and G we get dim ∆c ≤ 11. Theorem O shows that dim c < 16, and the dimension formula gives dim ∆ ≤ 26. Proof of (c) (1) Let dim ∆ ≥ 27. Then ∆ cannot be semi-simple by (a). This means that ∆ has a minimal commutative connected normal subgroup Ξ , and Ξ is either compact, (and then Ξ is contained in the center Z , see [1], 93.19), or Ξ is a vector group R , (and ∆ induces an irreducible representation on Ξ). The proof of (b) shows that Z is straight. The dual statement is also true. Z is not planar by (a), and Theorem B implies that Z is contained in a group ∆[a,W ] . As mentioned in the introduction, Z does not consist of elations, and a / ∈ W . This proves the first assertion of (c) . In particular, Ξ ≤ ∆[a,W ] if Ξ is compact. (2) Assume now that Ξ is a vector group and that Ξ|W 6= 1l. Choose z ∈ W such that z 6= z , and let c ∈ az \ {a, z} . The group Ξ induces on the orbit z a sharply transitive Lie group Ω ∼= Ξ/Ξz of dimension at most 8. Consider an element ω ∈ Ω which belongs to a unique one-parameter subgroup Π of Ω . Denote the connected component of ∆c∩Cs ω by Λ . Then Λ centralizes each element of Π and fixes z pointwise. Hence the fixed elements of Λ form a connected subplane FΛ . Moreover, Λ is a Lie group since Λ ∩ Z ≤ Zc = 1l, and dim Λ ≥ 27− dim c − dim Ω > 3 by Theorem O. The center Z acts effectively on FΛ because Z consists of homologies. If dimFΛ ≤ 4, then Z would be a Lie group by Theorem P. Therefore, FΛ is a Baer subplane, and we conclude from [1], 83.6 and 55.32(ii) that Λ is a compact Lie group of torus rank at most 1. Hence Λ ≤ SU2 and dim Λ ≤ 3. This contradiction proves that Ξ ≤ ∆[a,W ] as 90 Priwitzer, Salzmann asserted. If Ξ is not compact, then Ξ ∼= R by [1], 61.2. Together with the first part of (1) this implies that dim ∆/Cs Ξ ≤ 1. Proof of (d) (1) Whenever a 6= c / ∈ W , then ∆c is a Lie group because ∆c ∩ Z = 1l. If Λ denotes the stabilizer of a quadrangle and Φ = Λ ∩ Cs Ξ , then dim Λ/Φ ≤ 1 by the last remark in (c). Moreover, Φ is a Lie group, and the fixed elements of Φ form a ΞZ -invariant connected subplane F . Since Z acts effectively on F and Z is not a Lie group, it follows from Theorem P that F is a Baer subplane or F = P . Consequently, Φ is a compact Lie group of torus rank at most 1, and dim Φ ≤ 3. Thus, the existence of Ξ implies dim Λ ≤ 4. Letting ac ∩W = z , we conclude from Lemma O that dim cz < 8. (2) Assuming again that dim ∆ ≥ 27, we now study the action of ∆ on W . For v ⊆W and dim v = k > 0, the dimension formula [1], 96.10 and the last remarks in (1) imply 27 ≤ dim ∆ ≤ 3k+7+4 and k > 5. Similarly, if ∆ fixes a point z ∈W , then ∆ has only 8-dimensional orbits on W \ z , and ∆ is even doubly transitive on W \ z . In this case, the action of ∆v on v ≈ R is linear [1], 96.16(b), and the stabilizer Λ of a quadrangle has a connected subplane of fixed elements. With the arguments of (c) step (2), we would obtain dim Λ ≤ 3, but dim Λ ≥ 27−2 ·8−7 = 4. If, on the other hand, dim v = 8 for each v ∈W , then ∆ would be transitive on W ≈ S8 . Consequently, dim ∆ ≥ 36 by [1], 96. 19 and 23, and ∆ would be a Lie group, either by [1], 87.1 or by the dual of [1], 62.11. Hence there is some v ∈ W with dim v = k ∈ {6, 7} . Suppose that ∆ is doubly transitive on V = v . By results of Tits [1], 96. 16 and 17, either V is a sphere, or V is an affine or projective space and the stabilizer of two points fixes a real or complex line. In the latter case, the stabilizer Ω of three “collinear” points of V would have dimension at least 27−15, but the remarks at the end of (1) show that dim Ω ≤ 11. If V ≈ S6 , then ∆ has a subgroup Γ ∼= G2 , see [1], 96. 19 and 23, and ∆ would be a Lie group by Theorem G. Therefore, V ≈ S7 , and the Tits list [1], 96.17(b) shows that ∆ induces on V a group PSU5(C, 1) or PU3(H, 1). In the first case, ∆ contains a subgroup SU4C . As in the proof of (a) step (2), the element diag(1, 1,−1,−1) and its conjugates are Baer involutions. Two of these involutions fix a 4dimensional subplane F . The center Z acts effectively on F and hence would be a Lie group by Theorem P. In the only remaining case, ∆ has a subgroup Ψ which is locally isomorphic to U3(H, 1), compare [1], 94.27. Consider a maximal compact subgroup Φ of Ψ and its 10-dimensional factor Υ . From Lemma R we conclude that Υ ∼= U2H ∼= Spin5R and that the central involution σ of Υ is not planar. Thus, σ is a reflection [1], 55.29, and σ fixes only the points on the axis and the center. Moreover, the map Υ → PU3(H, 1) is injective and σ acts freely on V . Therefore, W is not the axis of σ , and σ fixes exactly two points on W . Hence Cs σ = ∇ is the stabilizer of a triangle. Let K be the connected component of the kernel ∆[V ] . Then dim ∆/K = dim U3(H, 1) = 21, and dim K ≥ 6. On the other hand, K acts effectively on the line av , and dim K ≤ 7 by Lemma O. Any representation of Ψ in dimension < 12 is trivial, see [1], 95.10. Therefore, Ψ induces the identity on the Lie algebra of K , and Ψ ◦ K = 1l. Consequently, KΦ ≤ ∇ , and dim∇ ≥ 6 + 13, but step (1) implies Priwitzer, Salzmann 91 dim∇ ≤ 2 ·7 + 4 = 18. This contradiction shows that ∆ is not doubly transitive on V . (3) Choose v ∈ W such that v = V has dimension < 8, and let c ∈ av \ {a, v} . The connected component Γ of ∆c is not transitive on V \ {v} and hence has an orbit u = U ⊂ V of dimension ≤ 6. By the last remarks in (1), we have dim Γ ≥ 13 and dimU ≥ 5. Consequently, Γ acts effectively on U . Assume that dimU = 6 and that Γ is doubly transitive on U . Step (1) implies that dim Γ ≤ 2 · 6 + 4 = 16, and Γ cannot be simple by [1], 96.17. From [1], 96.16 we conclude that U ≈ R and that Γu has a subgroup Φ ∼= SU3C . The representation of Φ on U ≈ C shows that each involution in Φ fixes a 2-dimensional subspace of U and so is planar. Two commuting involutions fix a 4-dimensional subplane, and Z would be a Lie group by Theorem P. Therefore, the connected component Ω of Γu has an orbit in U of dimension < 6. By step (1), we obtain dim Ω ≤ 9 and dim Γ ≤ 15. If ζ is in the center of Γ , and z 6= z ∈ U , then Γz = Γzζ fixes a quadrangle, and dim Γz ≤ 4 by step (1), but dim Γz ≥ 13− 6. Because Γ acts effectively on U , this shows that the center of Γ is trivial. Either Γ has a minimal normal subgroup X ∼= R , or Γ is a direct product of simple Lie groups, cp. [1], 94. 26 and 23. We will discuss the two possibilities separately in the next steps. (4) Let Γ be semi-simple. Any reflection α ∈ Γ has axis av , and α 6= α since Γ has trivial center. The set αα is contained in the connected component E of the elation group Γ[v,av] , and E is a normal subgroup of Γ . Hence E is itself a product of simple Lie groups, and E contains a non-trivial torus, but an involution is never an elation [1], 55.29. This contradiction shows that each involution in Γ is planar. Because dim Γ > 8, there exists a pair of commuting involutions. Their common fixed elements form a 4-dimensional subplane [1], 55.39, and Z would be a Lie group by Theorem P. Therefore, Γ cannot be semisimple. (5) We use the notation of (3) and determine the action of Ω on X . Note that u 6= u because Γ acts effectively on U . If u 6= z ∈ u , then z ⊂ u . By step (1), we have dim Ωz ≤ 4. From dim Γ ≥ 13 it follows that dim Ω ≥ 7 and hence 3 ≤ dim z ≤ dimu . The stabilizer Xu fixes each point of the connected subplane 〈a, c, uX〉 , and this subplane has dimension at least 8, since u is contained in a line and dimu > 2. From [1], 83.6 we infer that Xu is compact, and then Xu = 1l since X is a vector group. Because Ω acts linearly on X , the fixed elements of the connected component Λ of Ωz form a connected subplane F . As a group of homologies, the non-Lie group Z acts effectively on F , and Theorem P implies that F is a Baer subplane. As at the end of (c) step (2) it follows that Λ is isomorphic to a subgroup of SU2 . We know that dim Ω ≥ 7, and we conclude from (3) that there is a point z with dim z < 6. This gives dim Λ ≥ 2 and then Λ ∼= SU2 ∼= Spin3 . In particular, dim Λ = 3 and dim z ≥ 4. Therefore, any minimal Ω -invariant subgroup of X has dimension at least 4. Calling such a subgroup X from now on, we may assume that Ω acts irreducibly on X ∼= R , where 4 ≤ s ≤ 6. These three possibilities will be discussed in the last steps. Each case will lead to a contradiction. (6) The connected component Λ of Ωz acts reducibly on X by its very 92 Priwitzer, Salzmann definition. If s = 4, then Λ induces on X either the identity or a group SO3 . Each non-trivial orbit of Ω on X is 4-dimensional, and Ω is transitive on X\{1l} . In particular, Ω is not compact, and a maximal compact (connected) subgroup Φ of Ω has dimension at most 6. A theorem of Montgomery [1], 96.19 shows that Φ is transitive on the 3-sphere consisting of the rays in X ∼= R . Let r denote the ray determined by z . Then Λ ≤ Φr and Φ/Φr ≈ S3 . This implies dim Φ = 6 and dim Φr = 3. Moreover, Φr is connected by [1], 94.4(a), and hence Φr = Λ is simply connected. The exact homotopy sequence [1], 96.12 shows that Φ is also simply connected. Consequently, Φ ∼= Spin4 ∼= (Spin3)2 , compare [1], 94.31(c), and Φ contains exactly 3 involutions. If dimw = 6 for some w ∈ U , then w is open in U by [1], 96.11. Since w is also compact and U is connected, Φ would be transitive on U , but u = u . Hence, each stabilizer Φw has positive dimension and contains a (planar) involution γ . Let Fγ = {x ∈ W | x = x} . Then U is covered by the 3 sets Fγ , and these are homeomorphic to S4 . The sum theorem [1], 92.9 implies dimU ≤ 4, but we have seen at the beginning of (3) that dimU ≥ 5. This contradiction excludes the case s = 4. (7) If s = 5, then Ω acts effectively on X , and Ω′ is irreducible and simple, see [1], 95. 5 and 6(b). A table of irreducible representations [1], 95.10 shows that dim Ω′ ∈ {3, 10} , but we know from step (5) that 6 ≤ dim Ω′ ≤ 8. (8) Finally, let s = 6. Then Ω′ is semi-simple by [1], 95.6(b), and dim Ω′ ∈ {6, 8} . Note that SU2 ∼= Λ < Ω′ . Either Ω′ is even almost simple, or Ω′ has a factor Φ ∼= SU2 . By [1], 95.5, any Φ -invariant subspace of X has a dimension d dividing 6, but effective irreducible representations of SU2 exist only in dimensions 4k , compare [1], 95.10. Therefore, Ω′ is almost simple. The table [1], 95.10 shows that Ω′ must be one of the groups SO3C , SL3R , or SU3(C, r). The first two have no subgroup SU2 and can be discarded. The two unitary groups contain 3 diagonal involutions. Each one of these has an eigenvalue 1 and thus is planar. By [1], 55.39 their common fixed elements form a 4-dimensional subplane F . The center Z acts effectively on F , and Z would be a Lie group by Theorem P. This completes the proof of (d) and hence of Theorem L.