4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

Monatsh. Math. 136, 327±354 (2002) DOI 10.1007/s00605-001-0478-4

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups By

GuÈnter F. Steinke University of Canterbury, Christchurch, New Zealand Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002 # Springer-Verlag 2002 Abstract. This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, R). Various characterizations of this single Laguerre plane are obtained. 2000 Mathematics Subject Classi®cation: 51H15 Key words: Laguerre plane, elation group

1. Introduction A 4-dimensional Laguerre plane L ˆ …P; C; k† is an incidence structure consisting of a 4-dimensional locally compact point set P, a circle set C homeomorphic to R6 whose elements are subsets of P, called circles, and an equivalence relation k (parallelism) de®ned on the point set such that three mutually nonparallel points can be joined by a unique circle, such that the circles which touch a ®xed circle K at p 2 K partition Pnjpj (where jpj denotes the parallel class of p), such that each parallel class meets each circle in a unique point (parallel projection), and such that there is a circle that contains at least three points. Furthermore, the operations of joining three points by a circle, of intersecting two different circles, and touching are continuous with respect to the induced topologies on their respective domains of de®nition. For more information about topological Laguerre planes we refer to [6], [7] and [19]. Circles and parallel classes of 4-dimensional Laguerre planes are homeomorphic to the 2-sphere S2 and to R2 , respectively. Furthermore, the space  of all parallel classes can be identi®ed with a circle and thus is homeomorphic to S2 . The point set of a 4-dimensional Laguerre plane can be considered as a ®bre bundle over a circle with ®bres being the parallel classes. The projection map of the bundle is given by the parallel projection in the Laguerre plane. It was shown in [18] that the point space of a 4-dimensional Laguerre plane is bundleequivalent to the tangent bundle of the 2-sphere S2 . Furthermore, in a 4-dimensional Laguerre plane any two circles have at least one point in common, cf. [7, Satz 3.3.b] and [3, Satz 3.7.b].

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The classical (4-dimensional) complex Laguerre plane is obtained as the geometry of non-trivial plane sections of an elliptic cone in complex projective 3dimensional space with its vertex removed. In [16], we constructed the ®rst examples of non-classical 4-dimensional Laguerre planes. They were obtained by pasting together two halves of the classical complex Laguerre plane along a 3-dimensional separating set in the point set and are therefore called semi-classical planes; see 2.5 for an explicit description. These planes were characterized in [20] in terms of their automorphism groups and how they act on the sets of parallel classes and on the circle sets; see Theorem 2.5. In this paper we continue to study 4-dimensional Laguerre planes and determine all such planes admitting non-solvable automorphism groups and a full elation group, see below for a de®nition of elation group. 2. Four-Dimensional Elation Laguerre Planes The collection of all continuous automorphisms of a 4-dimensional Laguerre plane L forms a group with respect to composition, the automorphism group of L. This group is a Lie group with respect to the compact-open topology, see [4, Satz 3.9] or [14]. The kernel of a Laguerre plane consists of all automorphisms that ®x each parallel class. This collection of automorphisms is a closed normal subgroup of the automorphism group of the Laguerre plane. The maximum dimensions of the automorphism group and kernel of a 4dimensional Laguerre plane are 14 and 8, respectively. These dimensions are attained in the classical complex Laguerre plane: The kernel is isomorphic to 3 j C Cnf0g  and the connected component of the automorphism group of the classical 4-dimensional Laguerre plane that contains the identity is isomorphic 3 j C to …Cnf0g  SO3 …C††  (all collineations of the ambient complex projective 3-dimensional space that leave the elliptic cone and its vertex invariant). Every 4-dimensional Laguerre plane contains one further distinguished closed normal subgroup, the elation group
, which is the collection of all automorphisms in the kernel T that ®x no circle plus the identity. We obtain an elation Laguerre plane if
acts sharply transitively on the set of circles, see [17]. The automorphisms in this group induce elations in the associated Lie geometry and one obtains an elation generalized quadrangle; cf. [13] for generalized quadrangles and their relation to Laguerre planes and the other types of circle planes. The following characterizations of 4-dimensional elation Laguerre planes were obtained in [17], see also [10]. Theorem 2.1. For a 4-dimensional Laguerre plane L let
and T be the elation group and the kernel of L, respectively. Then the following are equivalent: (1) (2) (3) (4) (5) (6)

L is an elation Laguerre plane; the kernel T is transitive on the set of circles; T is at least 7-dimensional;
is 6-dimensional;
is connected and isomorphic to R6 ;
is sharply transitive on the set of circles.

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329

Associated with every point p of a 4-dimensional Laguerre plane L there is a derived incidence structure, called the derived af®ne plane Ap ˆ …Ap ; Lp † at p, whose point set Ap  R4 consists of all points of L that are not parallel to p and whose line set Lp consists of all restrictions to Ap of circles of L passing through p and of all parallel classes not passing through p. From the axioms of a Laguerre plane it readily follows that Ap is an af®ne plane. Indeed, each derived af®ne plane Ap of a 4-dimensional Laguerre plane is even a topological locally compact connected af®ne plane; see the coherence axioms in [6] and [7]. Moreover, Ap extends to a topological locally compact connected projective plane Pp , which we call the derived projective plane at p; compare [12, Corollary 43.7]. The axioms of a 4-dimensional Laguerre plane further imply that circles not passing through the distinguished point p induce ovals in Pp by removing the point parallel to p and adding in Pp the in®nite point of the lines that come from parallel classes of L. The line at in®nity of Pp (relative to Ap ) is a tangent to each of these ovals. We even obtain closed ovals in this way; compare [12, Proposition 55.18 and Theorem 55.11]. In Ap one has a parabolic curve. This gives a very convenient description of a Laguerre plane in one derived af®ne plane. One has the lines of the af®ne plane and a collection of parabolic curves. To obtain the entire Laguerre plane, however, one has to extend this model by one parallel class. The classical complex Laguerre plane can be characterized in various ways, cf. [4], [9], [15] and [20]. We give a summary of these characterizations. Theorem 2.2. A 4-dimensional Laguerre plane L is isomorphic to the classical complex Laguerre plane if and only if at least one of the following holds: (1) one derived af®ne plane is Desarguesian; (2) the automorphism group of L is at least 11-dimensional; (3) the kernel T of L is 8-dimensional. Assuming that L is even an elation Laguerre plane with automorphism group , then the classical complex Laguerre plane can be further characterized by each of the following: (4) (5) (6) (7)

is ¯ag-transitive; is transitive on the points of L; is transitive on the set of parallel classes of L; contains a compact group of dimension at least 2.

The beauty of elation Laguerre planes lies in the fact that each derived projective plane of an elation Laguerre plane is a dual translation plane with centre ! (the in®nite point of lines that come from parallel classes of the Laguerre plane). The stabilizer of a circle is linearly represented on the elation group
by conjugation. Therefore the well developed theory of translation planes, see [12], and, if the stabilizer of a circle is large enough, the representation theory of Lie groups can be applied to classify the most homogeneous elation Laguerre planes. This was carried out in [15] and all 4-dimensional elation Laguerre planes that admit 4dimensional stabilizers of a circle were classi®ed. We shall see that there is, up to isomorphism, a unique 4-dimensional elation Laguerre plane that admits an almost simple but non-simple Lie group of automorphisms.

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2.3. A matrix representation of 4-dimensional elation Laguerre planes. For each z 2 S2  R2 [ f1g let 0 1 A…z† D…z† ˆ @ B…z† A C…z† be a 6  2 matrix with 2  2 matrices A, B, C such that A…1† ˆ I2 and B…1† ˆ C…1† ˆ 0 where 0 and I2 denote the 2  2 zero and identity matrix, respectively, and such that for all z 2 R2 the second row of B(z) equals z and C…z† ˆ I2 . One can further assume that A…0† ˆ B…0† ˆ 0. Every 4-dimensional elation Laguerre plane can be represented by such a mapping D. Circles of the elation Laguerre plane L are of the form Kc ˆ f…z; c
D…z††jz 2 S2 g where c 2 R6 . The elation group
of L is given by all maps …z; w† 7! …z; w ‡ c
D…z†† for c 2 R6 ; the connected component of the kernel containing the identity consists of all maps …z; w† 7! …z; rw ‡ c
D…z†† for c 2 R6, r 2 R, r > 0. The continuity of the geometric operations in such a Laguerre plane described by a matrix valued mapping D reduces to the continuity of D in R2 and to limz!1 D…z†A…z† 1 ˆ D…1†, cf. [17, Proposition 5.8]. Note that a circle Kc passes through the in®nite point …1; …c1 ; c2 †† where c1 and c2 denote the ®rst and second component of the vector c 2 R6. Hence, circles through the point …1; 0† are of the form f…z; …c3 ; c4 †B…z† ‡ …c5 ; c6 ††jz 2 S2 g for c3 ; c4 ; c5 ; c6 2 R. Since this derived af®ne plane is a dual translation plane, one ®nds that fB…z†jz 2 R2 g is a spread set. It further follows that the circles that touch K0 at …1; 0† are of the form f…z; …c5 ; c6 ††jz 2 S2 g for c5 ; c6 2 R. In particular, the bundle of all circles through …1; 0† yields a 4-dimensional vector subspace of R6 and the bundle of all circles that touch K0 at …1; 0† yields a 2-dimensional vector subspace of R6 . This carries over to arbitrary bundles; compare [17, 4.5]. Also note that ff…z; …c3 ; c4 †B…z††jz 2 S2 gjc3 ; c4 2 Rg and ff…z; …c1 ; c2 †A…z††jz 2 S2 gjc1 ; c2 2 Rg represent the bundle of circles through …1; 0† and (0, 0) and the bundle of circles that touch K0 at (0, 0), respectively.

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331

2.4. Example. The semi-classical Laguerre planes from [16] can be described in the above fashion as follows. For 0 < p 4 1 and x; y 2 R let  2    y x x2 2xy y A…x; y† ˆ ; B…x; y† ˆ ; for y 5 0; 2 x2   2 2xy 2 y x y  py x x 2xy py B…x; y† ˆ ; for y 4 0: A…x; y† ˆ 2 2 ; x y 2pxy py x Every semi-classical Laguerre plane is isomorphic to an elation Laguerre plane as described above for exactly one p, 0 < p 4 1. One obtains the classical complex Laguerre plane for p ˆ 1. The automorphism group of Lp is 10-dimensional unless p ˆ 1. In any case, a semi-classical Laguerre plane admits the simple group PSL2 …R† as a group of automorphisms. In fact, the semi-classical Laguerre planes can be characterized among 4-dimensional elation Laguerre planes by this property, see [20, Theorem 5.8]. Theorem 2.5. A 4-dimensional elation Laguerre plane is semi-classical if and only if it admits a simple Lie group of automorphisms. 3. The SL2 …R†-Elation Laguerre Plane In this section L ˆ …P; C; k† always denotes a 4-dimensional elation Laguerre plane and , T and
 R6 denote its automorphism group, kernel and elation group, respectively. For a subgroup  of we denote by 1 the connected component of  containing the identity. We say that a subgroup  of is in the elation complement if its intersection with
is trivial, that is,  \
ˆ fidg. Since we can replace a subgroup in the elation complement by an isomorphic subgroup in the elation complement that ®xes a circle, cf. [20, Lemma 4.1], we only consider subgroups in the elation complement that ®x a circle. A subgroup  in the stabilizer of a circle K operates on the circle set C and on the elation group
by conjugation. The operation on
yields a linear and faithful representation of  on R6 . Furthermore, C and
can be identi®ed in such a way that the distinguished circle K corresponds to the identity. More precisely,  2
is identi®ed with …K†. Then both actions are equivalent, that is,  1 2
is identi®ed with …K† for  2 . In the matrix representation 2.4 of an elation Laguerre plane a circle L has coordinate vector c and this vector c also describes the corresponding element of
. We always make this identi®cation. Suppose that the automorphism group of a 4-dimensional elation Laguerre plane L is 10-dimensional. Let K be a circle of L and let  be the connected component of the group f 2 K jdet… † ˆ 1g that contains the identity. Then  is a closed, connected 3-dimensional subgroup of 1K and has discrete intersection with T. Furthermore, 1K is the semidirect product of  and … K \ T†1 and  is locally isomorphic to 1 =… 1 \ T†. A non-solvable Lie group contains a maximal semisimple subgroup (Levi complement). Furthermore, a semisimple Lie group is locally isomorphic to a direct product of almost simple (or locally simple) Lie groups, cf. [5, Theorem 19.12] or

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[12, 94.23]. We begin with a classi®cation of semisimple Lie groups that can occur as groups of automorphisms of elation Laguerre planes. Proposition 3.1. Let L be a 4-dimensional elation Laguerre plane and assume that L admits a connected semisimple Lie group  of automorphisms. Then  is in the elation complement and has discrete intersection with the kernel T. Furthermore,  is almost simple and isomorphic to PSL2 …C†, SO3 …R†, PSL2 …R† or SL2 …R†. Proof. Let  be a semisimple Lie group of automorphisms of L. Since
is an abelian normal subgroup of , we clearly see that  must be in the elation complement. Likewise, the connected component T1 of the kernel T is a solvable normal subgroup of , see [17, Corollary 3.6], so that  must have discrete intersection with T. A semisimple Lie group is at least 3-dimensional and, because dim T 5 7, the dimension of is at least 10. Hence either L is classical or dim  ˆ 3 by Theorem 2.2. In the former case,  is a subgroup of PSL2 …C† and thus isomorphic to PSL2 …C†, SO3 …R† or PSL2 …R†. In the latter case,  is almost simple and locally isomorphic to SO3 …R† or PSL2 …R†. Such groups have been investigated in [20] and Theorem 3.2 below has been proved. Therefore  is isomorphic to SO3 …R†, PSL2 …R† or SL2 …R†. & We give a summary of [20, Theorem 4.6]. Theorem 3.2. Let L be a 4-dimensional elation Laguerre plane that admits a non-solvable automorphism group. Then there is a circle K and a closed connected 3-dimensional subgroup  in the stabilizer of K such that  \ T is discrete and such that  acts ®xed-point-freely on K. (1) If  acts transitively on K, then  is isomorphic to SO3 …R†. (2) If  is not transitive on K but still acts ®xed-point-freely on K, then  is isomorphic to either PSL2 …R† or SL2 …R†. In this case,  has precisely three orbits on K, one 1-dimensional orbit O1 homeomorphic to the 1-sphere S1 and the two connected components of KnO1 . The effective operation of  on O1 is transitive and thus is equivalent to the standard operation of PSL2 …R† on S1 . In case   PSL2 …R†, the elation group splits into the direct sum of two 3-dimensional irreducible -invariant subspaces; in case   SL2 …R†, the group  acts irreducibly on
and the centre of  acts trivially on K. By Theorem 2.5 we obtain semi-classical Laguerre planes in the above situation unless   SL2 …R†. We investigate this case in the following and show that this leads to a single candidate for a Laguerre plane. In the next section we then verify the axioms of a 4-dimensional Laguerre plane for the incidence structure found in this section. 3.3. General assumptions for the remainder of this section. Let L be a 4dimensional elation Laguerre plane with elation group
 R6 . We represent L as in 2.3 by a matrix-valued map D. Let  be a closed connected subgroup of the

4-Dimensional Elation Laguerre Planes

automorphism group

333

of L such that

±   SL2 …R†; ±  ®xes the circle K0 ˆ f…z; 0†jz 2 S2 g; ±  has a 1-dimensional orbit O1  S1 in K0 and acts transitively on both connected components of K0 nO1 ; ±  acts irreducibly and faithfully on C 
 R6 ; ± the centre Z ˆ fI2 ; I2 g of  acts trivially on K0; ± the 1-dimensional orbit O1 contains the points …1; 0† and (0, 0); ± =Z acts on O1 like the group PSL2 …R† on S1 in its standard transitive    action; a b 1 ja; b; d 2 R; ad ˆ 1; a > 0 ; ± …1;0†  0 d    a 0 1 ± …0;0†  ja; c; d 2 R; ad ˆ 1; a > 0 . c d Up to equivalence there is a unique real linear irreducible faithful representation  of SL2 …R† on R6 . For example, one obtains such a representation  on the space of homogeneous polynomials of degree 5 in two variables X and Y by de®ning   a b for a; b; c; d 2 R, ad bc ˆ 1, to be the linear map that takes a homo c d geneous polynomial f (X, Y ) to the polynomial f …aX ‡ bY; cX ‡ dY†. With respect to the basis 5 4 3 2 2 3 4 5  X ; X Y; X Y ; X Y ; XY ; Y a b the following 6  6 matrix one obtains for  c d 0 5 a 5a4 b 10a3 b2 10a2 b3 B a4 c 3 2 2 a …ad ‡ 4bc† a b…4ad ‡ 6bc† ab …6ad ‡ 4bc† B B 3 2 B a c a2 c…2ad ‡ 3bc† a…a2 d 2 ‡ 6abcd ‡ 3b2 c2 † b…3a2 d 2 ‡ 6abcd ‡ b2 c2 † B 2 3 B a c ac2 …3ad ‡ 2bc† c…3a2 d 2 ‡ 6abcd ‡ b2 c2 † d…a2 d 2 ‡ 6abcd ‡ 3b2 c2 † B 4 @ ac c3 …4ad ‡ bc† c2 d…6ad ‡ 4bc† cd 2 …4ad ‡ 6bc† c5 5c4 d 10c3 d 2 10c2 d 3 1 b5 5ab4 b3 …4ad ‡ bc† b4 d C C C b2 d…3ad ‡ 2bc† b3 d 2 C 2 2 3C bd …2ad ‡ 3bc† b d C C d 3 …ad ‡ 4bc† bd4 A 5cd4 d5 We assume that 6 ± the action of  on Cis given  by , that  is, the  coordinate vector c 2 R of a a b a b circle Kc 2 C is taken by to c
 . c d c d

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Note that there is no loss of generality in making the above assumptions; we just have to coordinatise L suitably. It is readily seen that the subspace spanned by fe7 i ; . . . ; e6 g, i ˆ 1; . . . ; 6, is the only i-dimensional vector subspace of R6 which is invariant under all linear   1 b for b 2 R where e1 ; . . . ; e6 2 R6 are the standard basis transformations  0 1 vectors for R6, i.e., ei is the vector whose i-th entry and all whose other entries  is 1  1 b are 0. Since the automorphisms associated with for b 2 R ®x …1; 0†, they 0 1 leave invariant the 2-dimensional subspace C…1;0†;…1;0† of all circles that touch K0 at …1; 0† and the 4-dimensional subspace C…1;0† of all circles through …1; 0†; compare 2.3. Hence C…1;0†;…1;0† is spanned by e5 and e6 and C…1;0† is spanned by e3, e4, e5 and e6. One similarly ®nds that the 2-dimensional subspace C…0;0†;…0;0† of all circles that touch K0 at (0, 0) is spanned by e1 and e2 and the 4-dimensional subspace C…0;0† of all circles through (0, 0) is spanned by e1 ; e2 ; e3 and e4. Therefore C…1;0†;…0;0† ˆ C…1;0† \ C…0;0† is spanned by e3 and e4. This agrees with the standard interpretation of the matrices A(z) and B(z) in the matrix representation of a 4-dimensional elation Laguerre plane in terms of bundles of circles as given in 2.3. Finite points of L will be denoted by (z, w) with z; w 2 R2 or by (x, y, u, v) for x; y; u; v 2 R, where z ˆ …x; y† and w ˆ …u; v†, whichever is more convenient at the time. Likewise, we use c ˆ …c1 ; c2 ; c3 ; c4 ; c5 ; c6 † 2 R6 and c ˆ …c1 ; c2 ; c3 † with c1 ˆ …c1 ; c2 †; c2 ˆ …c3 ; c4 †; c3 ˆ …c5 ; c6 † 2 R2 simultaneously for a coef®cient vector c of a circle Kc. Then Kc ˆ f…z; c
D…z††jz 2 S2 g ˆ f…z; c1 A…z† ‡ c2 B…z† ‡ c3 †jz 2 R2 g [ f…1; c1 †g ˆ f…z; c1 f …z† ‡ c2 g…z† ‡ c3 h…z† ‡ c4 z ‡ …c5 ; c6 ††jz 2 R2 g [ f…1; …c1 ; c2 ††g where f(z) and g(z) denote the ®rst and second row of A(z), respectively, and h(z) denotes the ®rst row of B(z), that is, f(z), g(z) and h(z) are maps describing the circles Ke1 , Ke2 and Ke3 , respectively. The entries of h(z) will be denoted by uh(z) and vh(z), that is, h…z† ˆ …uh …z†; vh …z††, and likewise for f(z) and g(z). Note that the second row of B(z) equals z. Under the above general assumptions we can give an explicit formula for the action of  ˆ 1…1;0† on ®nite points. In order to distinguish the standard basis vectors for R6 from the standard basis vectors for R2, we denote the latter by e01 ˆ …1; 0† and e02 ˆ …0; 1†.   a b 2 SL2 …R† and let  2  be the associated autoLemma 3.4. Let L ˆ 0 d morphism of L. (Note that d ˆ a 1 .) Then  acts on the set of ®nite points R4 as follows:  : …z; w† 7! …d 3 zL

…2bd; b2 d 2 †; d 4 wL†:

4-Dimensional Elation Laguerre Planes

Proof. The point (z, w) lies on the circles K…0;0;w† 0 5 a 5a4 b 10a3 b2 10a2 b3 B0 a3 4a2 b 6ab2 B B0 0 a 3b …L† ˆ B B0 0 0 d B @0 0 0 0 0 0 0 0

335

and K…0;e02 ;w 5ab4 4b3 3b2 d 2bd2 d3 0

z† . Since 1

b5 b4 d C C b3 d2 C C; b2 d3 C C bd4 A d5

these circles are mapped under  to K…0;0;d4 wL† and K…0;de02 ;d4 …w z†L‡…2bd2 ;b2 d3 †† , respectively. The latter two circles have precisely one ®nite point (z0 , w0 ) in common. One ®nds w0 ˆ d 4 wL and dz0 ‡ d 4 …w z†L ‡ …2bd2 ; b2 d 3 † ˆ d 4 wL. From the latter equation one obtains z0 ˆ d 3 zL …2bd; b2 d 2 †. Thus (z, w) is mapped to …z0 ; w0 † ˆ …d 3 zL …2bd; b2 d 2 †; d 4 wL† under . & As an immediate consequence of the foregoing Lemma we can explicitly describe the orbits of  on the distinguished circle K0. Corollary 3.5. The 1-dimensional orbit O1 of  in K0 is    1 2 x ; 0; 0 x 2 R [ f…1; 0†g x; O1 ˆ 4 and the two 2-dimensional orbits of  in K0 are f…x; y; 0; 0†jx; y 2 R; 4y ‡ x2 > 0g and f…x; y; 0; 0†jx; y 2 R; 4y ‡ x2 < 0g: We are now able to obtain equations for the component functions of A(z) and B(z) for z 2 R4 with only a few parameters. Proposition 3.6. Let  ˆ …x; y† 2 f‡1; 0; 1g be the sign of 4y ‡ x2 and let ! 1 1 2 jy ‡ 14 x2 j 2 xjy ‡ 4 x j C…x; y† ˆ : 0 jy ‡ 14 x2 j3=2 Then the component functions of A(z) and B(z) satisfy the following identities:   3 2 3 1 3 x ; xy ‡ x ; h…x; y† ˆ h…0; †C…x; y† ‡ 4 2 8 1=2     1 2 1 3 3 2 3 4 x ; x y‡ x ; g…x; y† ˆ g…0; † y ‡ x ‡ 2xh…0; † C…x; y† ‡ 4 2 2 16 1=2   1 5 1 5 f …x; y† ˆ f …0; † y ‡ x2 ‡ xg…0; † y ‡ x2 ‡ x2 h…0; † C…x; y† 4 2 4 2   5 4 5 3 3 x ; x y ‡ x5 : ‡ 16 4 16

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Proof. We ®rst consider the ®rst row h(z) of B(z).The corresponding circle is  a b . Using Lemma 3.4 Ke3 . This circle is mapped to K…0;0;a;3b;3b2 d;b3 d2 † under  0 d we thus obtain a functional  equation h…z† ˆ h…x; y† where z ˆ The point  for   …x; y†. a b a b 3 2 2 4 …2bd; b d †; d h…z† , which …z; h…z†† 2 Ke3 is mapped to d z 0 d 0 d must be on K…0;0;a;3b;3b2 d;b3 d2 † . Hence       a b a b 4 3 2 2 d h…z† ˆ ah d z …2bd; b d † 0 d 0 d     a b …2bd; b2 d2 † ‡ 3b d 3 z 0 d ‡ …3b2 d; b3 d 2 † for all z 2 R2 , a; R; ad ˆ 1; a > 0. Given z ˆ …x; y† we let a ˆ d ˆ 1 and b; d 2 
a b x 3 …2bd; b2 d 2 † ˆ 0; y ‡ 14 x2 and b ˆ 2. Then d z 0 d        1 b 1 1 1 h…x; y† ˆ h 0; y ‡ x2 ‡ 3b 0; y ‡ x2 ‡ …3b2 ; b3 † 4 4 0 1     x 1 1 3 2 3 1 2 x ; xy ‡ x3 : ˆ h 0; y ‡ x2 ‡ 4 4 2 8 0 1 We now let x ˆ b ˆ 0. Then

  a 0 ˆ ah…0; d 4 y† d4 h…0; y† 0 d

and thus

 4 d h…0; d y† ˆ h…0; y† 0 4

0 d6



for all y; d 2 R; d > 0. Combining these two equations, we obtain the equation for h(x, y) in the proposition. Likewise, one obtains the equations for g(z) and f(z) by considering the  circles  a b Ke2 and Ke1 , respectively. Ke2 is mapped to K…0;a3 ;4a2 b;6ab2 ;4b3 ;b4 d† under  . 0 d Therefore       a b a b 4 3 3 2 2 ˆa g d z …2bd; b d † d g…z† 0 d 0 d     a b 2 3 2 2 ‡ 4a bh d z …2bd; b d † 0 d     a b 2 3 2 2 …2bd; b d † ‡ 6ab d z 0 d ‡ …4b3 ; b4 d†

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for all z 2 R2 , a; b; d 2 R, ad ˆ 1, a > 0. For a ˆ d ˆ 1 and b ˆ 2x we then ®nd      x 1 1 1 2 g…x; y† ˆ g 0; y ‡ x2 ‡ 2xh 0; y ‡ x2 4 4 0 1   1 3 3 2 3 x ; x y ‡ x4 ‡ 2 2 16 for all x; y 2 R and b ˆ x ˆ 0 yields

 6 d g…0; d y† ˆ g…0; y† 0 4

0 d8



for all y; d 2 R; d > 0. Taking the two equations together we obtain the stated   identity for g(x, y). a b . Finally, Ke1 is mapped to K…a5 ;5a4 b;10a3 b2 ;10a2 b3 ;5ab4 ;b5 † under  0 d Therefore ! !   a b a b 4 5 3 2 2 d f …z† ˆa f d z …2bd; b d † 0 d 0 d !   a b 4 3 2 2 ‡ 5a bg d z …2bd; b d † 0 d !   a b 3 2 3 2 2 ‡ 10a b h d z …2bd; b d † 0 d !   a b 2 3 3 2 2 ‡ 10a b d z …2bd; b d † 0 d ‡ …5ab4 ; b5 † for all z 2 R2 , a; b; d 2 R, ad ˆ 1, a > 0. For a ˆ d ˆ 1 and b ˆ 2x we then ®nd        x 1 1 2 5 1 2 5 2 1 2 2 f …x; y† ˆ f 0; y ‡ x ‡ xg 0; y ‡ x ‡ x h 0; y ‡ x 4 2 4 2 4 0 1   5 4 5 3 3 x ; x y ‡ x5 ‡ 16 4 16 for all x; y 2 R and b ˆ x ˆ 0 yields

 8 d f …0; d y† ˆ f …0; y† 0 4

0 d 10



for all y; d 2 R; d > 0. Taking the two equations together we obtain the stated identity for f(x, y). & The proposition shows that the circles of the Laguerre plane L are uniquely determined by the values of f …0;  1†, g…0;  1† and h…0;  1†. In the following proposition we obtain some restrictions on the values of h…0;  1† and show that f …0;  1† and g…0;  1† are in fact determined by h…0;  1†.

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Proposition 3.7. Let  ˆ  1 and let h…0; † ˆ …uh ; vh †, g…0; † ˆ …ug ; vg † and f …0; † ˆ …uf ; vf †. Then uh > 0 and uh 5 3: Furthermore, ug …9

uh † ˆ 8uh vh ; vg ˆ uh ‡

uf …6

uh † ˆ 5uh vg ;

vf ˆ ug ‡

9vh ug 8uh 6vh uf 5uh

; :

In particular, f …0; † and g…0; † are completely determined by h…0; † unless ‡1 u‡1 h ˆ 9 or uh ˆ 6.   h…z† for z 2 R2 form a spread set. Proof. Recall that the matrices B…z† ˆ z Hence B…x; y† B…0; † must be regular for all …x; y† 6ˆ …0; †. In particular, ! vh 2t3 3t2 uh 2 det…B…2t; t † B…0; †† ˆ det 2t  t2 ˆ t4 ‡ …uh

3†t2 ‡ 2vh t ‡ uh

must be non-zero for all t 2 R. This implies uh > 0. Similarly, det…B…2t; 

2

t †

B…0; †† ˆ det

3t2

2t3

3†t

!

t2

2t

ˆ t4 ‡ 2…uh

…uh

3†t2

must be non-zero for all t 2 R. This implies uh 5 3. The maps f …x; y†, g…x; y† and h…x; y† are differentiable in x and y at all points (x, y) for which 4y ‡ x2 6ˆ 0. Hence, if two circles touch at a point, then their describing functions must have the same derivatives at the point; cf. [17, Remark 5.11.a]. To apply this criterion we ®rst ®nd the partial derivatives of f(x, y), g(x, y) and h(x, y) at …0; †.  We haveC…0; † ˆ I2 and the of C…x; y† at …0; †  partial1 derivatives   0 0 2 . One then ®nds with the and Cx …0; † ˆ are Cy …0; † ˆ 0 32  0 0 product rule   2 0 ˆ 12 h…0; † ˆ 12 …2uh ; 3vh †; hy …0; † ˆ h…0; †Cy …0; † 0 3 ˆ 12 …0; 3 uh †; hx …0; † ˆ h…0; †Cx …0; † ‡ …0; 32 †   3 0 ˆ 12 …3ug ; 4vg †; gy …0; † ˆ g…0; †Cy …0; † ‡ 12 g…0; † ˆ 12 g…0; † 0 4 gx …0; † ˆ g…0; †Cx …0; † ‡ 2h…0; † ˆ …2uh ; 2vh 12 ug †;   4 0 1 ˆ 2 f …0; † ˆ 12 …4uf ; 5vf †; fy …0; † ˆ f …0; †Cy …0; † ‡ f …0; † 0 5 ˆ …52 ug ; 52 vg 12 uf †: fx …0; † ˆ f …0; †Cx …0; † ‡ 52 g…0; †

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We now consider the circle Ke2 described by the map g(z). Let K…0;r;s† , r; s 2 R2 , be the circle through …1; 0† that touches Ke2 at ……0; †; g…0; ††. Then g(z) and rB…z† ‡ s have the same derivative at …0; †. Therefore rBy …0; † ˆ gy …0; † and rBx …0; † ˆ gx …0; †. From the above we ®nd that   3  uh 2 vh : By …0; † ˆ 0 1 Since uh > 0, we see that By …0; † is regular. Hence r ˆ gy …0; †By …0; † therefore

1

and

gy …0; †By …0; † 1 Bx …0; † ˆ gx …0; †: Now 1 By …0; † Bx …0; † ˆ  2uh 1



3vh 2uh

3

 uh : 0

Thus 8…uh †2 ˆ …8uh vg 2…4vh

ug †uh ˆ 3…3

9vh ug † uh †ug

The second equation yields ug …9

uh † ˆ 8uh vh

and the ®rst equation yields the expression for vg. One likewise obtains that fy …0; †By …0; † 1 Bx …0; † ˆ fx …0; † and one ®nds 5uh vg ˆ …5uh vf …5vg

uf †uh ˆ 2…3

6vh †uf ; uh †uf :

From these two equations the respective identities for uf and vf follow. uh ˆ 9 or uh ˆ 6 can only occur if  ˆ ‡1 since uh 5 3. In all other cases we can solve the equations for ug and uf . Substituting the values for ug and vg into the formulae for uf and vf , we see that ug , vg , uf , and vf are completely determined by uh and vh . & The derived af®ne plane at …1; 0† is a dual translation plane. Dualising this plane we obtain a (projective) translation plane P. Since the group  induces a group of collineations of P, the collineation group P is at least 7-dimensional. P cannot be Desarguesian as this implies that L is classical. So the collineation group of P is either 8- or 7-dimensional. All 4-dimensional translation planes admitting collineation groups of dimension 7 or 8 have been classi®ed by Betten, see [12, x73] for an overview. If the collineation group of P is 7-dimensional, then P is isomorphic to one of the planes in [2, Satz 5].

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So far we havenot the full group . It is well  known that   SL2 …R† is  used  a b ja; b; d 2R; ad ˆ 1; a > 0 and the automorphism  generated by   0 d 0 1 associated with the matrix . In order to know how  acts on L it 1 0 suf®ces to know the action  of .  0 1 Lemma 3.8. Let S ˆ and let  2  be the associated automorphism 1 0 of L. Then  acts on the ®nite points (z, w) with z 6ˆ 0 as follows:  : …z; w† 7! …h…z†A…z† 1 S; wA…z† 1 S†: Proof. The point …z; w†; z 6ˆ 0, is on the circle K…r;0;0† as well as on the circle K…s;e1 ;0† where r ˆ wA…z† 1 and s ˆ …w h…z††A…z† 1 . Both circles also have the point (0, 0) in common. Since  1 ˆ 1…0;0† , the point (0, 0) must be mapped to …1; 0† under . Hence we ®nd the image of (z, w) as 0 the ®nite point 1 of intersection 0 0 S of the images of K…r;0;0† and K…s;e01 ;0† . Since …S† ˆ @ 0 S 0 A, one ®nds that S 0 0 K…r;0;0† is mapped to K…0;0;rS† and K…s;e01 ;0† is mapped to K…0;e02 ;sS† . These two circles intersect in …1; 0† and …h…z†A…z† 1 S; wA…z† 1 S†. & We can now put the pieces together and show that there is at most one candidate for the kind of Laguerre plane we are investigating in this section. Theorem 3.9. There is at most one elation Laguerre plane admitting SL2 …R† as a group of automorphisms. In this plane one has f …0; 1† ˆ …125; 0†; g…0; 1† ˆ …0; 5†; h…0; 1† ˆ …5; 0†; f …0; 1† ˆ …1; 0†; g…0; 1† ˆ …0; 1†; h…0; 1† ˆ … 1; 0†; that is,

   1 2 2 h…x; y† ˆ 5y ‡ 2x ; x y ‡ x ; 2     3 2 1 2 2 x g…x; y† ˆ x…10y ‡ 3x †; 5y ‡ x y ; 2 2   2 2   3 2 3 2 f …x; y† ˆ 5 5y ‡ x ; ; 2x 5y ‡ x 2 2

for 4y ‡ x2 5 0, and h…x; y† ˆ …y ‡ x2 ; xy†; g…x; y† ˆ …x…2y ‡ x2 †; y…y ‡ x2 ††; f …x; y† ˆ …y2 ‡ 3x2 y ‡ x4 ; xy…2y ‡ x2 ††; for 4y ‡ x2 4 0.

4-Dimensional Elation Laguerre Planes

341

0

1 0 0 S Proof. Since …S† ˆ @ 0 S 0 A, the circle Ke4 is mapped under  to K e3 . S 0 0 Hence …h…z†A…z† 1 S; zA…z† 1 S† ˆ …z; z† 2 K e3 for all z 2 R2 , z 6ˆ 0 and therefore h…h…z†A…z† 1 S† ˆ

zA…z† 1 S

for all z 2 R2 , z 6ˆ 0. We use this identity for points z ˆ …2t;  t2 † for t 2 R. The ®rst components of both sides when multiplied by a suitable function give us polynomials in t. Comparing corresponding coef®cients and using Proposition 3.7 we shall be able to determine h…0; † ˆ …uh ; vh † and thus g…0; † ˆ …ug ; vg †   and f …0; † ˆ …uf ; vf †. 1 t 2 that One ®nds from Proposition 3.6 with C…2t;  t † ˆ 0 1 h…2t; 

t2 † ˆ …uh ‡ 3t2 ; vh ‡ …3

g…2t; 

t2 † ˆ …ug ‡ 4uh t ‡ 4t3 ; vg ‡ …4vh

f …2t; 

t2 † ˆ …uf ‡ 5ug t ‡ 10uh t2 ‡ 5t4 ; vf ‡ …5vg

uh †t

uf †t ‡ 5…2vh

2t3 †; ug †t ‡ 2…3

2uh †t2

3t4 †;

ug †t2 ‡ 10…

uh †t3

4t5 †:

One further computes h…2t; 

t2 †A…2t; 

1 t2 † 1 S ˆ …; † d

where t2 †

d ˆ det A…2t;  ˆ uf vg

ug vf ‡ 4…uf vh

‡ 4…5ug ‡ 2…3uh  ˆ uh vf

‡ 3…2

15vg ‡ 20uh †t4 ‡ 4…ug

uh vg ††t2 5vh †t5

5†t6 ‡ t8 ;

uf vh ‡ …5…uh vg

‡ …15vg  ˆ uh vg

vf †t3 ‡ …uf

uh vf †t ‡ 2…3uf ‡ 5…ug vh

uf

ug vh uh †t4

ug vh †

3uf †t ‡ 3…vf

20uh †t3 ‡ 5…5vh 3ug t ‡ 3…vg

5ug †t2

ug †t4 ‡ 3…5

2uh †t2 ‡ …8vh

3uh †t5

2t7 ;

ug †t3

t6 :

Now the ®rst component of h…d ; d† is 4d1 2 …3 2 ‡ uh j4d ‡  2 j† which must be equal to the ®rst component of …2t;  t2 †A…2t;  t2 † 1 S, that is, 1 1 …3 2 ‡ j4d ‡  2 j† ˆ …uf ‡ …5ug 2vf †t ‡ …uf 10vg ‡ 10uh †t2 2 4d d ‡ 5…ug 4vh †t3 ‡ 5…2uh 3†t4 ‡ 3t6 † for all t 2 R. Note that 4d ‡  2 must have the same sign as  so that j4d ‡  2 j ˆ …4d ‡  2 †. Multiplying both sides by 4d 2 we obtain a polynomial

342

G. F. Steinke

equation in t. Expanding we obtain that F …t† ˆ 4……5 ‡ 3uh †vh ‡ 2‰3…5

2ug †t11 ‡




uh †…uh vf

uf vh †…uh vg

uf …3uh ‡ †…uh vf 2…uf vg ‡ …uh vf

ug vh †

uf vh †

ug vf †…3uh ug ‡ 5ug

uf vh †2 …3 ‡ uh †

4…uf

2vf †Št uh …uh vg

ug vh ††…uf vg

ug vf †

ˆ0 for all t 2 R. Hence each coef®cient of F …t† must be 0. Equating the coef®cient of t11 to 0 we ®nd ug ˆ 12 …5 ‡ 3uh †vh . From Proposition 3.7 we obtain …9

uh †…5 ‡ 3uh †vh ˆ 16uh vh :

Therefore vh ˆ 0

or …uh

5†…uh ‡ 3† ˆ 0:

Since uh > 0, we have uh ‡ 3 6ˆ 0. Furthermore, because uh 4 3, the case uh ˆ 5 can only occur when  ˆ ‡1. Hence vh ˆ 0

or u‡1 h ˆ 5:

We ®rst assume that vh ˆ 0. Then ug ˆ 0 and vf ˆ 0; moreover, vg ˆ uh and uf ˆ 5…uh †2 6 uh

from Proposition 3.7. Substituting these expressions into the constant term of

F …t† and equating to 0 we obtain 0 ˆ …uh †2

Since

uh

6uh ‡ 5 ˆ …uh

3 ‡ 2†…uh

3

2†:

5 3 by Proposition 3.7, we ®nd that uh ˆ 3 ‡ 2; ug ˆ 0;

vh ˆ 0; vg ˆ 3 ‡ 2;

uf ˆ 63 ‡ 62;

vf ˆ 0:

This gives us the stated values for f …0;  1†, g…0;  1† and h…0;  1†. We ®nally assume that u‡1 h ˆ 5. Then ‡1 u‡1 g ˆ 10vh ;

‡1 2 25 u‡1 f ˆ 4 …20 ‡ 9…vh † †;

‡1 2 1 v‡1 g ˆ 4 …20 ‡ 9…vh † †;

‡1 2 1 ‡1 v‡1 f ˆ 2 vh …80 ‡ 27…vh † †

from Proposition 3.7. Substituting these expressions into the coef®cient of t1 of F …t† and equating to 0 we obtain ‡1 2 ‡1 4 v‡1 h …27…vh † ‡ 80†…9…vh †

2 40…v‡1 h † ‡ 2000† ˆ 0:

Hence v‡1 h ˆ 0 and we ®nd the same values for f(0, 1), g(0, 1) and h(0, 1) as before. &

4-Dimensional Elation Laguerre Planes

343

Remark 3.10. The circle describing maps h, g and f can be given in closed form as follows:   3 2 1 3 1 2 x ‡ y ‡ x …2; x†; h…x; y† ˆ 3y ‡ x ; 2 4 4     3 4 1 3 2 x ‡ y ‡ x2 4x; 2y x ; g…x; y† ˆ 2x…3y ‡ x2 †; 3y2 8 4 2    49 29 9 f …x; y† ˆ 63y2 ‡ 39x2 y ‡ x4 ; x 24y2 ‡ x2 y ‡ x4 8 2 4   1 41 ‡ y ‡ x2 62y ‡ x2 ; x…26y ‡ 9x2 † : 4 2 It is interesting to note that the terms not involving jy ‡ 14 x2 j in the above formulae describe an incidence structure that almost is an elation Laguerre plane. More precisely, let   3 2 1 3 ~ x ; h…x; y† ˆ 3y ‡ x ; 2 4   3 4 2 2 ~ x ; g…x; y† ˆ 2x…3y ‡ x †; 3y 8    ~f …x; y† ˆ 63y2 ‡ 39x2 y ‡ 49 x4 ; x 24y2 ‡ 29 x2 y ‡ 9 x4 : 8 2 4 Then Kc ˆ f…z; c1~f …z† ‡ c2 ~g…z† ‡ c3 ~h…z† ‡ c4 z ‡ …c5 ; c6 ††jz 2 R2 g [ f…1; …c1 ; c2 ††g de®nes a set of circles for c ˆ …c1 ; c2 ; c3 ; c4 ; c5 ; c6 † 2 R6. The derived incidence structure at …1; 0† is a dual translation plane; in fact, this af®ne plane A is isomorphic to the dual of Betten's translation plane from [1, Satz 2.b]. The circle Ke2 induces an oval in A. However, Ke1 does not induce a topological oval in A, since, for example, the equation ~f …x; y† ˆ …0; 1† has no real solution whereas by [3, Satz 3.1] a topological oval in a 4-dimensional projective plane possesses no exterior lines. In fact, the substructure obtained for all circles Kc with c1 ˆ 0 is isomorphic to the corresponding substructure obtained from the single 4-dimensional elation Laguerre plane found in [15, x8]. 4. The Axioms of a Laguerre Plane In this section L ˆ …P; C; k† always denotes the incidence structure we obtained in Theorem 3.9, that is, P ˆ …R2 [ f1g†  R2 and C ˆ ff…z; c1 A…z† ‡ c2 B…z† ‡ c3 †jz 2 R2 g [ f…1; c1 †gjc1 ; c2 ; c3 2 R2 g

344

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where

 A…x; y† ˆ

y2 ‡ 3x2 y ‡ x4

x…2y ‡ x2 †   y ‡ x2 xy ; B…x; y† ˆ x y

for 4y ‡ x2 4 0, and A…x; y† ˆ B…x; y† ˆ


2 5 5y ‡ 32 x2 x…10y ‡ 3x2 † 5y ‡ 2x2 x

xy…2y ‡ x2 † y…y ‡ x2 †

 ;


2 ! 2x 5y ‡ 32 x2

; 5y ‡ 32 x2 y 12 x2
! x y ‡ 12 x2 ; y

for 4y ‡ x2 5 0. Later on it will prove convenient to rewrite the matrices A(x, y) and B(x, y) as follows ! 2xq2 5q2 ; A…x; y† ˆ 2xq 15 q…q 4x2 † ! 1 2 q ‡ 12 x2 x…q ‡ x † 5 ; B…x; y† ˆ 1 3 2 x 5q 10 x where q ˆ 5y ‡ 32 x2 and 4y ‡ x2 5 0. We verify the axioms of a topological Laguerre plane, that is, we show ± that three mutually non-parallel points can uniquely be joined by a circle, ± that a circle K and a pair (p, q) of non-parallel points with p 2 K determine a unique circle that touches K at p and passes through q, and ± that the geometric operations are continuous. We begin with a few useful observations, one of which shows that circles look like `parabolae' on the subset f…x; y; u; v† 2 R4 j4y ‡ x2 < 0g. Lemma 4.1. The matrix-valued functions A(z) and B(z) are continuous on R2 . Let z ˆ …x; y† 2 R2 . Then  4 y; for 4y ‡ x2 4 0;
det A…z† ˆ 3 2 4 5y ‡ 2 x ; for 4y ‡ x2 5 0: In particular, A(z) is invertible for all z 2 R2 , z 6ˆ 0. Furthermore, A…z† ˆ B…z†2 for 4y ‡ x2 4 0. We now come to two homogeneity properties of L which allow us later in the veri®cation of the axioms to specialise some of the points involved. A straightforward computation shows that the formulae in Lemmata 3.4 and 3.8 indeed de®ne automorphisms of L where  from Lemma 3.8 is extended by …0; w† ˆ …1; wS† and …1; w† ˆ …0; wS†. Hence we obtain the following.

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345

Proposition 4.2.   SL2 …R† acts as a group of automorphisms of L and has the three orbits f…x; y; 0; 0† 2 R4 j4y ‡ x2 > 0g; f…x; y; 0; 0† 2 R4 j4y ‡ x2 < 0g and f…x; y; 0; 0† 2 R4 j4y ‡ x2 ˆ 0g [ f…1; 0; 0†g j
has three orbits in the point space, two 4-dimenin K0. Correspondingly,   sional orbits

4 2 O‡ 4 ˆ f…x; y; u; v† 2 R j4y ‡ x > 0g;

O4 ˆ f…x; y; u; v† 2 R4 j4y ‡ x2 < 0g and one 3-dimensional orbit O3 ˆ f…x; y; u; v† 2 R4 j4y ‡ x2 ˆ 0g [ f1gR2 : Furthermore, the action of  on C is given by . By Proposition 4.2 every point in O4 can be mapped to the point …0; 1; 0; 0† by some automorphism. The stabilizer of this point allows us some further reduction. Lemma 4.3. Let subgroup corresponding to the   SO2 …R† be the canonical  cos t sin t rotation group t 2 R . Then  has the ®xed points sin
t cos t …0; 1; 0; 0† and 0; 15 ; 0; 0 in K0. Furthermore, every point of K0 can be mapped under  to a point …0; r; 0; 0† for some r 2 R.   0 1 is an element of . Proof. The automorphism  belonging to S ˆ 1 0 Using the formula from 3.8 we explicitly ®nd for the action of  on K0
8 x 1 ; y ; 0; 0 ; for 4y ‡ x2 4 0; …x; y† 6ˆ …0; 0†; > y > >
> 4y < 2x 2 10y‡3x2 ; …10y‡3x2 †2 ; 0; 0 ; for 4y ‡ x 5 0; …x; y† 6ˆ …0; 0†;  : …x; y; 0; 0† 7! > > …1; 0; 0†; for x ˆ y ˆ 0; > > : …0; 0; 0; 0†; for …x; y† ˆ 1:
1 It now readily follows that …0; 1; 0; 0† and 0; 5 ; 0; 0 are the only ®xed points of  on K0. Since  is abelian and connected, both points must be ®xed by every element of . By [11, 6.7.1] the group SO2 …R† can act on K0  S2 only trivially or with two ®xed points. Furthermore, in the latter case, every 1-dimensional orbit
separates the two ®xed points. Therefore …0; 1; 0; 0† and 0; 15 ; 0; 0 are the only ®xed points of . Since f…0; r; 0; 0†jr 2 R; 1 4 r 4 1g is a segment connecting both points, we obtain that every orbit of  in K0 must intersect this segment, that is, every orbit contains a point …0; r; 0; 0† for some r 2 R. & Proposition 4.4. The derived incidence structure Ap of L at each point p in the 3-dimensional orbit O3 ˆ f…x; y; u; v† 2 R4 j4y ‡ x2 ˆ 0g [ f1gR2 is a

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topological af®ne plane. In particular, B…z1 † z1 ; z2 2 R2 , z1 6ˆ z2 .

B…z2 † is invertible for all

Proof. By Proposition 4.2 it suf®ces to verify the derived incidence  structure  1 0 at pˆ …1; 0† is a  topological af®ne plane. In this case, 0 2 2 0 t t 2 3 B…z† z 2 R , where B…z† denotes the transpose of B(z), is one of 0 1 the spread sets described in [2, Satz 5] ± in the notation of [2] it is the one with parameters w ˆ p ˆ 0, z ˆ 35, q ˆ 3. Hence fB…z†jz 2 R2 g is a spread set and Ap is an af®ne plane, in fact, a dual translation plane, cf. 2.3. Furthermore, with respect to the induced topologies, Ap is a topological 4-dimensional af®ne plane by [2, Satz 5]. Finally, B…z1 † B…z2 † is invertible for all z1 ; z2 2 R2 , z1 6ˆ z2 , by the de®nition of a spread set. & Proposition 4.5. Three mutually non-parallel points of L can be uniquely joined by a circle. Proof. Let …zj ; wj †, j ˆ 1; 2; 3, three mutually non-parallel points. In view of Proposition 4.4 we can assume that none of the points is in the 3-dimensional orbit O3; in particular, zj 6ˆ 1. The coef®cient vector c ˆ …c1 ; c2 ; c3 † 2 R6 of a circle Kc that passes through the three points satis®es the system of linear equations 0 1 A…z1 † A…z2 † A…z3 † c
@ B…z1 † B…z2 † B…z3 † A ˆ …w1 ; w2 ; w3 †: I2 I2 I2 We show that the coef®cient matrix 0

1 A…z1 † A…z2 † A…z3 † D…z1 ; z2 ; z3 † ˆ @ B…z1 † B…z2 † B…z3 † A I2 I2 I2

is invertible. This then implies the existence and uniqueness of the circle joining the three points. We subtract the third column from the ®rst and second columns and then subtract …B…z2 † B…z3 †† 1 …B…z1 † B…z3 †† times the second column (where this column is multiplied by the matrix on the right) from the ®rst column. After these elementary column operations we obtain the matrix 0 1 C…z1 ; z2 ; z3 † A…z2 † A…z3 † A…z3 † @ 0 B…z2 † B…z3 † B…z3 † A 0 0 I2 where C…z1 ; z2 ; z3 † ˆ A…z1 †

A…z3 †

…A…z2 †

A…z3 ††…B…z2 †

B…z3 †† 1 …B…z1 †

Hence det D…z1 ; z2 ; z3 † ˆ det C…z1 ; z2 ; z3 †det…B…z2 †

B…z3 ††

B…z3 ††:

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347

and D…z1 ; z2 ; z3 † is invertible if and only if C…z1 ; z2 ; z3 † is invertible by Proposition 4.4. We distinguish four cases according to how the points are distributed among the two 4-dimensional orbits O‡ 4 and O4 . Case 1. All three points are in O4 , that is, 4yj ‡ x2j < 0 for j ˆ 1, 2, 3. By Proposition 4.1 we may assume that z3 ˆ …0; 1†. Then A…z3 † ˆ I2 and B…z3 † ˆ I2 . Using Lemma 4.4 we then have C…z1 ; z2 ; z3 † ˆ B…z1 †2 ˆ …B…z1 †

I2 …B…z2 †2 I2 †…B…z2 † ‡ I2 † 1 …B…z1 † ‡ I2 † B…z2 ††…B…z1 † ‡ I2 †

since B2 I2 ˆ …B I2 †…B ‡ I2 † for any 2  2 matrix B. Hence D…z1 ; z2 ; z3 † is invertible by Proposition 4.4. 2 Case 2. Two points are in O4 and one in O‡ 4 , say 4yj ‡ xj < 0 for j ˆ 2, 3 and 2 4y1 ‡ x1 > 0. By Proposition 4.2 and Lemma 4.3 we may assume that z3 ˆ …0; 1† and z2 ˆ …0; r† for some r > 0, r 6ˆ 1. Then A…z3 † ˆ I2 , B…z3 † ˆ I2 , A…z2 † ˆ r2 I2 and B…z2 † ˆ rI2 . Hence

C…z1 ; z2 ; z3 † ˆ A…z1 †

I2 ‡ …1 ‡ r†…B…z1 † ‡ I2 †

ˆ A…z1 † ‡ rI2 ‡ …1 ‡ r†B…z1 †: Let q1 ˆ 5y1 ‡

3 2 2 x1 .

Then  2 1 r‡1 x41 ‡ ……q1 det C…z1 ; z2 ; z3 † ˆ 5 2

r†2 ‡ r…q1

1†2 †x21

 ‡ …5q21 ‡ …r ‡ 1†q1 ‡ r†…q21 ‡ …r ‡ 1†q1 ‡ 5r†

>0 since q1 ˆ 5…y1 ‡ 14 x21 † ‡ 14 x21 > 14 x21 5 0 and r > 0. Hence D…z1 ; z2 ; z3 † is invertible. 2 Case 3. Two points are in O‡ 4 and one in O4 , say 4yj ‡ xj > 0 for j ˆ 2, 3 and
2 4y1 ‡ x1 < 0. By
Proposition 4.2 and Lemma 4.3 we may assume that z3 ˆ 0; 15 1. Then A…z3 † and B…z3 † are the diagonal and z2 ˆ 0; 15 r for some  r > 0, r 6ˆ 5 0 1 0 and D1 ˆ , respectively, and A…z2 † ˆ r 2 D5 matrices D5 ˆ 0 15 0 15 and B…z2 † ˆ rD1 . Hence

C…z1 ; z2 ; z3 † ˆ A…z1 †

D5

ˆ A…z1 † ‡ rD5

…1 ‡ r†D5 D1 1 …B…z1 †

D1 †

…1 ‡ r†D5 D1 1 B…z1 †

and 1 det C…z1 ; z2 ; z3 † ˆ …rx41 …5…r ‡ 1†y21 28ry1 ‡ 5r…r ‡ 1††x21 5 ‡ …y21 5…r ‡ 1†y1 ‡ 5r†…5y21 5…r ‡ 1†y1 ‡ r††:

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G. F. Steinke

We consider this expression as a polynomial in x21 , that is, we de®ne p…t† ˆ rt2

…r…r ‡ 1†y21 ‡

…y21

28ry1 ‡ 5r…r ‡ 1††t

5…r ‡ 1†y1 ‡ 5r†…5y21

5…r ‡ 1†y1 ‡ r†

for t 2 R. The assumption 4y1 ‡ x21 < 0 yields 0 4 x21 < Evaluating p(t) and its derivative at 4y1 we ®nd p… 4y1 † ˆ 5……4y21 ‡ …y1

1†2 †r2

p0 … 4y1 † ˆ

4y2 ‡ 1†r ‡ y22 † < 0:

5…r 2 ‡ …y22

4y1 ; in particular, y1 < 0.

2y1 …y1 ‡ 1†2 r ‡ y21 ……y1

1†2 ‡ 4†† > 0;

Since p(t) describes a convex parabola, we obtain p…t† > 0 for all t < det C…z1 ; z2 ; z3 † > 0 and D…z1 ; z2 ; z3 † is invertible.

4y1 . Hence

2 Case 4. All three points are in O‡ 4
, that is, 4yj ‡ xj
> 0 for j ˆ 1, 2, 3. As in 1 1 case 3, we may assume that z3 ˆ 0; 5 and z2 ˆ 0; 5 r for some r > 0, r 6ˆ 1, so that again

C…z1 ; z2 ; z3 † ˆ A…z1 † ‡ rD5 But now det C…z1 ; z2 ; z3 † ˆ



r‡1 2

2

x41 ‡ ……q1

…1 ‡ r†D5 D1 1 B…z1 †:

r†2 ‡ r…q1

1†2 †x21 ‡ …q1

r†2 …q1

1†2

where q1 ˆ 5y1 ‡ 32 x21 . Since each term in this sum is non-negative, we obtain that det C…z1 ; z2 ; z3 † 5 0. Moreover, det C…z1 ; z2 ; z3 † ˆ 0 implies x1 ˆ 0 and either q1 ˆ 1 or q1 ˆ r. However, this yields z1 ˆ z3 or z1 ˆ z2, respectively, ± a contradiction. Hence det C…z1 ; z2 ; z3 † > 0 and D…z1 ; z2 ; z3 † is invertible. This shows that D…z1 ; z2 ; z3 † is invertible in any case and therefore the three points can be uniquely joined by a circle. & We now come to tangent circles. In [10] the concept of a (dual) pseudo-oval was introduced. L owen showed that a 4-dimensional compact dual pseudo-oval is equivalent to a 4-dimensional elation Laguerre plane, see also [8]. The secant condition of such a dual pseudo-oval corresponds to the axiom of joining in the associated Laguerre plane and the tangent condition corresponds to the axiom of touching. Each 4-dimensional compact dual pseudo-oval can be represented by a matrix-valued map D as in 2.3, called an admissible parametrization. If D is an admissible parametrization such that the associated dual pseudo-oval OD satis®es the secant condition and which is `smooth' at a point z 2 S2 , then OD also satis®es the tangent condition at z, see [10, Proposition 4.9]. Here smooth means that D is differentiable at z and each directional derivative of D at z in direction h 2 R2 n f…0; 0†g has rank 2 and the span of this 6  2 matrix is independent of h. This condition can be veri®ed for the map D we are using here and all z ˆ …x; y† 2 R2 , 4y ‡ x2 6ˆ 0. Note that the tangent condition is satis®ed at points where D is not smooth by Proposition 4.4, but see also [10, 4.12]. Using the correspondence between dual pseudo-ovals and elation Laguerre planes the existence of tangent circles easily follows. However, because the correspondence between the tangent

4-Dimensional Elation Laguerre Planes

349

condition of a dual pseudo-oval and the axiom of touching in the Laguerre plane occurs only implicitly in [10], we exhibit the direct proof without the use of dual pseudo-ovals. It will also show how to explicitly ®nd a tangent circle. Note that in [17, 5.10] we obtained a necessary condition for two circles in a 4dimensional elation Laguerre plane to touch in points where B(z) is differentiable. In fact, this condition also proves to be necessary and suf®cient in our situation even though we do not yet know we have a Laguerre plane.
Lemma 4.6. Let z0 ˆ …0; 1† or z0 ˆ 0; 15 and let e02 ˆ …0; 1†. A circle Kc through …1; e02 † and …z0 ; 0† touches the circle K0 at …z0 ; 0† if and only
if
c ˆ …0; 1; 0; 2; 0; 1† or c ˆ 0; 1; 0; 2; 0; 15 for z0 ˆ …0; 1† or z0 ˆ 0; 15 , respectively. Proof. It readily follows that one obtains tangent circles for the stated values of c, or see the proof of the following Lemma 4.7. In order to prove that c must have the stated form we consider the following map f : S2 ! S2 de®ned by 8 > e0 …A…z† A…z0 ††…B…z† B…z0 †† 1 ; for z 2 R2 ; z 6ˆ z0 ; > < 2 for z ˆ z0 ˆ …0;
1†; f …z† ˆ …0; 2†; > …0; 2†; for z ˆ z0 ˆ 0; 15 ; > : 1; for z ˆ 1: f(z) is de®ned in such a way that that the circle with coef®cient vector …e02 ; f …z†; e02 A…z0 †

f …z†B…z0 ††

…1; e02 †,

…z0 ; 0† and …z; 0† for z 6ˆ z0 and that for z ˆ z0 the is the circle through above vector yields the stated c. From this description and the fact that we have a tangent circle for the particular c it follows that f is injective on S2 . Furthermore, f clearly is continuous on S2 nf1; z0 g. We now verify the continuity of f at the remaining poins 1 and z0 . Case 1. z0 ˆ …0; 1†. Then A…z0 † ˆ I2 and B…z0 † ˆ I2 . For z ˆ …x; y† with 4y ‡ x2 4 0, z 6ˆ z0 , we ®nd that f …z† ˆ e02 …B…z† I2 † ˆ e02 z by Lemma 4.1. Hence f is continuous at z0. Moreover, f(z) tends to 1 for z ! 1, 4y ‡ x2 4 0. For 4y ‡ x2 5 0 one computes  …q ‡ 1†…q ‡ 5† ‡ q…x2 ‡ 4† ; f …z† ˆ 4x 4…q ‡ 1†…q ‡ 5† ‡ x2 …x2 ‡ 4†  10 ‡ x2 …3q ‡ 5† 2q2 2…q ‡ 1† ; 4…q ‡ 1†…q ‡ 5† ‡ x2 …x2 ‡ 4† where q ˆ 5y ‡ 32 x2 . Note that q 5 14 x2 5 0. We write f(z) as (u, v). Then   …x2 ‡ 4†…4q 4x2 † 5 jxj juj ˆ jxj 1 ‡ 4…q ‡ 1†…q ‡ 5† ‡ x2 …x2 ‡ 4† and f(z) tends to 1 if x ! 1. If x is bounded but z ! 1, then q must go to 1 and so does v in this case. Hence f(z) tends to 1 for z ! 1, 4y ‡ x2 5 0. Together with the above result we obtain that f is continuous at 1.

350

G. F. Steinke

0; 15






5 0

0





1 0

0



. Then A…z0 † ˆ and B…z0 † ˆ and one Case 2. z0 ˆ 1 1 5 5 computes 8  2 2 2 8q‡1†x2 2…q 1†2 …q‡1† > ; 2 …3q 4…q ; for 4y ‡ x2 5 0; 4x 4…q …q1†21†‡x‡qx < 2 2 …x2 ‡4† 2 2 1† ‡x …x ‡4† f …z† ˆ   > …y 1†…5y2 1† x2 …5y‡1† 5y2 2y‡1 x2 : ; for 4y ‡ x2 4 0 ; x …5y 2 2 1†…y 1† x …5y 1†…y 1† x where again q ˆ 5y ‡ 32 x2 . Note that 4…q 1†2 ‡ x2 …x2 ‡ 4† ˆ 0 if and only if z ˆ z0 and …5y 1†…y 1† x2 ˆ 4y2 ‡ …y 1†2 …4y ‡ x2 † 5 1 for 4y ‡ x2 4 0. We write f(z) as (u, v). It is easy to see that juj > 13 jxj as long as q > 12 in case 4y ‡ x2 4 0. Thus f(z) tends to 1 if x goes to 1. (Note that q 5 14 x2 so that q ! 1 if x ! 1.) If x is bounded but z ! 1, then q and y must go to 1 and so does v. Finally, the continuity of f at z0 follows from juj 4 2jxj and jv ‡ 2j 4 jq 1j ‡ 2x2 for (q, x) suf®ciently close to (1, 0). Hence f is injective and continuous on S2 in any case. The Brouwer theorem on the invariance of domain shows that f must be a homeomorphism of S2 . Thus, every circle through …1; e02 † and …z0 ; 0† except one has a second point of intersection 6ˆ …z0 ; 0† with K0. The distinguished circle that intersects K0 in exactly one point is obtained precisely for the stated coef®cient vector c. & Lemma 4.7. Let z0 ˆ …0; y0 † where either y0 ˆ 1 or y0 ˆ 15. A
circle Kc touches the circle K0 at …z0 ; 0† if and only if c ˆ a
I2 jy20 j B…z0 † A…z0 † for some a 2 R2 . Proof. We ®rst verify that for the stated coef®cient vectors c we indeed obtain tangent circles at …z0 ; 0† where z0 ˆ …0; 1† or z0 ˆ …0; 15†. The circle Kc has the form     2 Kc ˆ B…z0 †B…z† ‡ A…z0 † z 2 R2 [ f…1; a†g: z; a
A…z† jy0 j It readily follows that Kc passes through …z0 ; 0†. Furthermore, Kc ˆ K0 if a ˆ 0. Let F…z† ˆ A…z†

2 B…z0 †B…z† ‡ A…z0 †: jy0 j

Then …z0 ; 0† is the only point of intersection of Kc with K0 for a 6ˆ 0 if F(z) is 2 invertible for all
z 2 R , z 6ˆ z0 . We separately verify this condition for z0 ˆ …0; 1† 1 and z0 ˆ 0; 5 . Case 1. z0 ˆ …0; 1†. For the determinant of F(z), z ˆ …x; y†, one ®nds  2 for 4y ‡ x2 4 0; ……x ‡ …y ‡ 1†2 †2 ; det F…z† ˆ 1 2 2 2 4 1† † ‡ 4…q ‡ 1† †; for 4y ‡ x2 5 0; 5 ……x ‡ …q where q ˆ 5y ‡ 32 x2 . Clearly, det F…z† 5 0 for 4y ‡ x2 4 0, and det F…z† ˆ 0 in this case if and only if x ˆ 0 and y ˆ 1, i.e. z ˆ z0 . Furthermore,

4-Dimensional Elation Laguerre Planes

det F…z† 5 z ˆ z0 .

4 5

351

> 0 for 4y ‡ x2 5 0, because q 5 0. Hence F(z) is invertible unless

Case 2. z0 ˆ …0; 1†. For the determinant of F(z), z ˆ …x; y†, one ®nds (1 1†2 4y x2 †2 4…y 1†4 ; for 4y ‡ x2 4 0; 5 …5…y det F…z† ˆ for 4y ‡ x2 5 0; …x2 ‡ …q 1†2 †2 ; where q ˆ 5y ‡ 32 x2 . Clearly, det F…z† 5 0 for 4y ‡ x2 5 0, and det F…z† ˆ 0 in this case if and only if x ˆ 0 and q ˆ 1, i.e. z ˆ z0. For 4y ‡ x2 4 0 we have 1 …5…y 5

1†2

4y

x2 †2

4…y

1 1†4 5 …5…y 5 ˆ …y

1†2 †2

4…y

1†4

1†4

5 1: Hence F(z) is invertible unless z ˆ z0 . Conversely, let Kc, c 6ˆ 0, be a tangent circle to K0 at …z0 ; 0†. Using the rotation group  from Lemma 4.3 we can transform Kc into a tangent circle whose coef®cient vector has ®rst entry 0. After applying a homothety hr : …z; w† 7! …z; rw†, r 2 R, r 6ˆ 0, we may further assume that the second entry of the coef®cient vector is 1. This shows that we can map Kc by a rotation and a homothety to circle Kc0 that 0 touches K0 at the same point …z0 ; 0† and passes through …1;
e2 †. By Lemma 4.6 we 0 0 1 then
have c ˆ …0; 1; 0; 2; 0; 1† or c ˆ 0; 1; 0; 2; 0; 5 for z0 ˆ …0; 1† or z0 ˆ 0; 15 , respectively. In
both cases c0 can be written as c0 ˆ e02
E where E ˆ I2 jy20 j B…z0 † A…z0 † . Under a homothety hr this  coef®cient vectorc0 is taken to rc0 ˆ re02
E; under a cos t sin t the vector rc0 is taken to …re02 Rt †
rotation associated with Rt ˆ sin t cos t
E for z0 ˆ …0; 1† and to
r sin t cos4 t
2cos2 t sin2 t‡ 15 sin4 t ; r cos t …cos4 t 2cos2 t sin2 t ‡ 5 sin4 t†
E for z0 ˆ 0; 15 . Hence c must be of the stated form. & Proposition 4.8. For each circle K and non-parallel points p and q such that p 2 K there is a unique circle through q that touches K at p. Proof. Let p ˆ …z0 ; w0 † and q ˆ …z1 ; w1 † be two non-parallel points and let K ˆ Kd , d 2 R6 , be a circle through p. Since the elation group
is transitive on C, we may assume that d ˆ 0 and w0 ˆ 0. We also know by Proposition 4.4 that the derived incidence structure at a point of the 3-dimensional orbit O3 is an af®ne plane so that we may further assume that z0 ˆ …x0 ; y0 † 2 R2 , 4y0 ‡ x20 6ˆ 0. Using the group , Proposition 4.2 ®nally allows us to restrict ourselves to z0 ˆ …0; 1†
and z0 ˆ 0; 15 . We suppose that Kc is a circle through q that touches K0 at p. By
Lemma 4.7 the coef®cient vector c is of the form c ˆ a
I2 jy20 j B…z0 † A…z0 † for some a 2 R2 . We then just have to determine for which such c the circle Kc passes

352

G. F. Steinke

through q, that is, we solve the linear equation  w0 ˆ a
I 2

0

if z1 ˆ …x1 ; y1 † 2 R2 and 

2 jy0 j B…z0 †

w0 ˆ a
I 2

1

 C A…z0 † B @ B…z1 † A I2  2 B…z0 †B…z1 † ‡ A…z0 † jy0 j

2 jy0 j B…z0 †

 ˆ a
A…z1 †

A…z1 †

1 I2 A…z0 † @ 0 A ˆ a 0 

0

if z1 ˆ 1. In the latter case, there clearly is a unique a 2 R2 and the tangent circle is unique in this case. In the former case, A…z1 † jy20 j B…z0 †B…z1 † ‡ A…z0 † ˆ F…z1 † is invertible as seen in the proof of Lemma 4.7. Therefore a is uniquely determined in this case too. This proves that Kc exists and is unique. & Since, obviously, each circle intersects each parallel class in L, Propositions 4.5 and 4.8 can be summarized as follows. Theorem 4.9. L is a Laguerre plane. It remains to show that the geometric operations are continuous on their respective domains of de®nition. By [17, Proposition 5.8] the veri®cation of continuity can be reduced in our situation to two simple convergence properties involving the matrices A(z) and B(z). Lemma 4.10. A…z† to 0.

1

and B…z†A…z†

1

converge to the zero matrix as z tends

Proof. Since A(z) and B(z) are continuous matrix-valued functions of z, each entry of A…z† 1 and B…z†A…z† 1 , z 6ˆ 0, also depends continuously on z. Let N ˆ f…x; y† 2 R2 jx2 ‡ jyj ˆ 1g: Then N is a compact subset of R2 not containing 0. Thus, each entry of A…z† 1 and B…z†A…z† 1 is bounded on N. p
For z ˆ …x; y† ˆ 6 …0; 0† let t ˆ x2 ‡ jyj. Then t > 0 and we de®ne z1 ˆ xt ; ty2 so that z1 2 N. Now A(z) and B(z) can be found from A…z1 † and B…z1 †, respectively, by  where T ˆ

t 0

 0 . Thus 1

A…z† ˆ t4 TA…z1 †T

1

;

2

1

;

B…z† ˆ t TB…z1 †T

A…z†

1

ˆ t 4 TA…z1 † 1 T

B…z†A…z†

1

ˆ t 2 TB…z1 †A…z1 † 1 T

1

; 1

:

4-Dimensional Elation Laguerre Planes

Since each entry of A…z1 † 1 and B…z1 †A…z1 † B…z†A…z† 1 converge to 0 as t tends to 1.

1

is bounded, the entries of A…z†

353 1

and &

Theorem 4.9 and Lemma 4.10 show that the assumptions of [17, Proposition 5.8] are satis®ed and we therefore ®nally obtain the following result. Theorem 4.11. L is a topological 4-dimensional Laguerre plane. 5. Characterizations We conclude this paper with some characterizations of the SL2 …R†-elation Laguerre plane constructed in Section 3 and the semi-classical Laguerre planes in terms of their automorphism groups. Theorem 5.1. A 4-dimensional elation Laguerre plane L with automorphism group is isomorphic to the SL2 …R†-elation Laguerre plane if and only if one of the following holds. (1) There is a circle K and a closed, connected 3-dimensional group  4 that acts irreducibly on C; (2) L admits SL2 …R† as a group of automorphisms.

K

Proof. The SL2 …R†-elation Laguerre plane obviously satis®es (1) and (2). Furthermore, as seen in Section 3, (2) gives rise to the SL2 …R†-elation Laguerre plane. Let  be a group as in (1). Since  acts irreducibly on C  R6 , it cannot be solvable. Therefore,  must be almost simple and must have discrete intersection with the kernel. From [17, Theorem 4.6] it now follows that   SL2 …R† and we have (2). & Theorem 5.2. Let L be a 4-dimensional elation Laguerre plane with automorphism group . Then the following statements are equivalent. (1) ®xes no parallel class and is at least 10-dimensional; (2) L admits a non-solvable Lie group of automorphisms; (3) L admits a Lie group of automorphisms locally isomorphic to PSL2 …R†; (4) L is semi-classical or isomorphic to the SL2 …R†-elation Laguerre plane. Proof. A semi-classical Laguerre plane admits a group of automorphisms isomorphic to PSL2 …R† in the stabilizer of the circle K0;0;0 . Furthermore, the action of PSL2 …R† on K0;0;0 and thus on the set  of parallel classes is equivalent to the standard action of PSL2 …R† as a subgroup of PSL2 …R† on S2  C [ f1g. By construction, the SL2 …R†-elation Laguerre plane admits SL2 …R† as a group of automorphisms and this group ®xes no parallel class. Both kinds of planes have an automorphism group of dimension at least 10. This shows that the semi-classical Laguerre planes and the SL2 …R†-elation Laguerre plane have all the properties stated in the theorem. Conversely, (2) implies (4); this follows from Proposition 3.1, Theorem 2.5, Theorem 5.1 and the fact that if is non-solvable, then it must contain a semisimple Lie group of automorphisms. Furthermore, Theorem 3.2 shows that (1) implies (3) and, clearly, (3) implies (2). &

354

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References [1] Betten D (1973) 4-dimensionale Translationsebenen mit 8-dimensionaler Kollineationsgruppe. Geom Dedicata 2: 327±339 [2] Betten D (1975) 4-dimensionale Translationsebenen mit genau einer Fixrichtung. Geom Dedicata 3: 405±440 [3] Buchanan T, Hahl H, L owen R (1980) Topologische Ovale. Geom Dedicata 9: 401±424 [4] Fortsch A (1982) Kollineationsgruppen endlichdimensionaler topologischer Benzebenen. Dissertation, Erlangen [5] Freudenthal H, de Vries H (1969) Linear Lie Groups. New York: Academic Press [6] Groh H (1968) Topologische Laguerreebenen I. Abh Math Sem Univ Hamburg 32: 216±231 [7] Groh H (1970) Topologische Laguerreebenen II. Abh Math Sem Univ Hamburg 34: 11±21 [8] Joswig M (1999) Pseudo-ovals, elation Laguerre planes, and translation generalized quadrangles. Beitr Algebra Geom 40: 141±152 [9] Lowen R (1981) Projectivities and the geometric structure of topological planes. In: Plaumann P, Strambach K (eds) Geometry ± von Staudt's Point of View, pp. 339±372. Dordrecht: Reidel [10] Lowen R (1994) Topological pseudo-ovals, elation Laguerre planes, and elation generalized quadrangles. Math Z 216: 347±369 [11] Montgomery D, Zippin L (1955) Topological Transformation Groups. New York: Wiley Interscience [12] Salzmann H, Betten D, Grundh ofer T, Hahl H, L owen R, Stroppel M (1995) Compact Projective Planes. Berlin: de Gruyter [13] Schroth AE (1995) Topological Circle Planes and Topological Quadrangles. Essex, UK: Longman [14] Steinke GF (1986) The automorphism group of Laguerre planes. Geom Dedicata 21: 55±58 [15] Steinke GF (1987) Eine Klassi®kation 4-dimensionaler Laguerre-Ebenen mit groûer Automorphismengruppe. Habilitationsschrift, Kiel [16] Steinke GF (1990) Semiclassical 4-dimensional Laguerre planes. Forum Math 2: 233±247 [17] Steinke GF (1991) The elation group of a 4-dimensional Laguerre plane. Monatsh Math 111: 207±231 [18] Steinke GF (1991) The point space of 4-dimensional Laguerre planes. Arch Math 56: 100±104 [19] Steinke GF (1995) Topological circle geometries. In: Buekenhout (ed) Handbook of Incidence Geometry, pp. 1325±1354. Amsterdam: Elsevier [20] Steinke GF (1999) On 4-dimensional elation Laguerre planes admitting simple Lie groups of automorphisms. Forum Math 11: 79±103 Author's address: Department of Mathematics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand