Special Issue (2003), S153–S157
Advances in Geometry
( de Gruyter 2003
16-dimensional compact projective planes with 3 fixed points
Helmut Salzmann
Dedicated to Professor Adriano Barlotti on the occasion of his 80th birthday
Let P ¼ ðP; LÞ be a topological projective plane with a compact point set P of finite
(covering) dimension d ¼ dim P > 0. A systematic treatment of such planes can be
found in the book Compact Projective Planes [15]. Each line L A L is homotopy
equivalent to a sphere Sl with l j 8, and d ¼ 2l, see [15] (54.11). In all known examples, L is in fact homeomorphic to Sl . Taken with the compact-open topology, the
automorphism group S ¼ Aut P (of all continuous collineations) is a locally compact
transformation group of P with a countable basis, the dimension dim S is finite [15]
(44.3 and 83.2).
The classical examples are the planes PK over the three locally compact, connected
fields K with l ¼ dim K and the 16-dimensional Moufang plane O ¼ PO over the
octonion algebra O. If P is a classical plane, then Aut P is an almost simple Lie
group of dimension Cl , where C1 ¼ 8, C2 ¼ 16, C4 ¼ 35, and C8 ¼ 78.
In all other cases, dim S c 12 Cl þ 1 c 5l. Planes with a group of dimension su‰ciently close to 12 Cl can be described explicitly. More precisely,
the classification program seeks to determine all pairs ðP; DÞ, where D is a connected
closed subgroup of Aut P and bl c dim D c 5l for a suitable bound bl d 4l 1.
This has been accomplished for l c 2 and also for b4 ¼ 17. Here, the case l ¼ 8
will be studied; the value of bl varies with the configuration of the fixed elements
of D.
Most theorems that have been obtained so far require additional assumptions on
the structure of D. If dim D d 27, then D is always a Lie group [12].
By the structure theory of Lie groups, there are 3 possibilities: (i) D is semi-simple,
or (ii) D contains a central torus subgroup, or (iii) D has a minimal normal vector
subgroup, cf. [15] (94.26). The first two cases are understood fairly well:
(a) If D is semi-simple and dim D > 28, then D G SL 3 H and P is a Hughes plane
(as described in [15] §86), or D G Spin 9 ðR; rÞ with r c 1, or P G O, see [10], [11].
(b) If D contains a central torus, and if dim D > 30, then D 0 G SL 3 H, see [13].
A group D of type (iii) fixes a point or a line, cf. [3] (XI.10.19). Hence (a) and (b)
imply
S154
Helmut Salzmann
(c) If dim D > 30 and D has no fixed element, then P is a Hughes plane or
P G O.
The case that D fixes exactly one element has been treated in [14]:
(d) If dim D d 35 and if D fixes one line and no point, then P is a translation plane.
All such planes have been determined in [6], [7], [9]. Either P G O or dim D ¼ 35.
Little progress has been made in the cases where D fixes exactly two elements,
necessarily a point and a line. If dim D d 40, then P and its dual are translation
planes [15] (87.7). All translation planes with dim D d 38 are described in [15] (82.28).
(e) If dim D d 34 and D fixes exactly 2 points and only one line, then D contains a
translation group of dimension at least 15.
(f ) If dim D d 33 and D fixes 2 points and 2 lines, then D contains a translation group
T G R8 and a compact subgroup F G Spin8 R.
A method to construct all planes with exactly 2 fixed points have been given in [8].
A smaller dimension of D su‰ces if D fixes more than two points (the last case to
be considered):
Theorem. If dim D d 32 and D has (at least) 3 fixed points, then D contains a transitive
translation group T. Either dim D ¼ 32 and a maximal semi-simple subgroup C of D is
isomorphic to SU4 C, or dim D d 37 and P G O.
Translation planes with a group C G SU4 C have already been studied in [5].
Examples of proper translation planes such that TC has a fixed point set S A S2 are
given in [6].
According to the sti¤ness result [15] (83.23), the stabilizer L of a non-degenerate
quadrangle satisfies dim L c 14. The proof of the theorem depends decisively on
Bödi’s improvement [1] of the sti¤ness theorem:
(j) If the fixed elements of the connected Lie group L form a connected subplane
E, then L is isomorphic to the 14-dimensional compact group G2 or its subgroup
SU3 C or dim L < 8. If E is a Baer subplane (dim E ¼ 8), then L is a subgroup of
SU2 C.
Corollary. From dim L > 8 it follows that dim E ¼ 2.
Proof. Assume that dim E ¼ 4. If L is any line of E and if c A LnE, then dim L c > 0
and the fixed elements of L c form a Baer subplane hE; ci. Hence dim L c c 3 and
dim L c 11. An alternative proof is given by [15] (96.35).
r
Proof of the Theorem. 1) For any closed subgroup G c D and any point x the
dimension formula dim G ¼ dim Gx þ dim x G holds, see [15] (96.10). This fact will be
used repeatedly without mention.
2) By the sti¤ness theorem, the stabilizer ‘ of a triangle satisfies dim ‘ c 30. Hence
16-dimensional compact projective planes with 3 fixed points
S155
all fixed points of D are incident with the same line W . There are at least 3 fixed
points u; v; w A W and the sti¤ness theorem implies dim D c 38.
3) Because of results (a) and (b), the group D has a minimal normal subgroup
Y G Rt . Choose a B W and % A P c Y such that P G R and a % 0 a. Since D acts
linearly on Y, the centralizer Cs % is also the centralizer of P, and the dimension
formula gives dim Cs P d 32 t. The connected component L of D a V Cs P fixes the
orbit a P pointwise, and the fixed elements of L form a connected subplane E, see [15]
(42.1). By (j) we have dim D a t c dim L c 14 and t d 2; moreover, dim L ¼ 14
or dim L c 8.
4) Assume first that t < 8. Then L G G2 is compact. Remember that the action
of any compact or semi-simple Lie group on a real vector space is completely
reducible ([2] (35.4)). Each irreducible module of G2 on R16 has a dimension divisible
by 7, see [15] (95.10). Since P L ¼ P, it follows from t c 7 that the commutator
½L; Y is trivial.
5) The last statement implies that the orbit a Y is contained in E. Because Y is
commutative, Ya fixes each point of a Y . Hence Ya acts trivially on the subplane E
generated by a P and u; v; w, and the connected component of Ya is contained in L,
but L is simple and L V Y ¼ 1. Therefore, dim Ya ¼ 0 and dim a Y ¼ t ¼ 2.
6) Denote the connected component of D a by ‘. From steps 3) and
pffiffiffiffi 5) it follows that
dim ‘ ¼ 16. Consequently, ‘ has a 2-dimensional radical P ¼ ‘, and ½L; P ¼ 1.
Hence E P ¼ E. If c is a point of E and c A awnfa; wg, then dim Pc > 0. On the other
hand, Pc acts trivially on the smallest closed subplane containing a; c; u; v, and this
subplane coincides with E by [15] (32.7); thus the connected component of Pc would
belong to the simple group L. This contradiction shows that t d 8.
7) If t ¼ 8, then 16 c dim ‘ ¼ dim %‘ þ dim L c t þ 14 ¼ 22 and dim L d 8. Consider the smallest closed subplane F containing a Y and u; v; w, and assume that
P 0 F ¼ F‘ . Then ‘ induces on F a group ‘=K of dimension c7, see [15] (83.17).
Hence dim K d 9 and K contains G2 . The Corollary implies that dim F ¼ 2 and
then dim ‘=K c 1 and dim K > 14. This contradiction shows F ¼ P and Ya ¼ 1
(because Ya fixes F pointwise). By (j) there are two possibilities: either L G G2 for
some % A Y, or L G SU3 C for each choice of %, and ‘ acts transitively on Ynf1g by
[15] (96.11). These cases will be treated separately.
8) Suppose that L G G2 and that L is contained in the maximal semi-simple subgroup C of D. By minimality of Y and [15] (95.6b), the group C acts irreducibly on Y
and L < C. Cli¤ord’s Lemma [15] (95.5) implies that L cannot be contained in a
proper factor of C, hence C is almost simple. Inspection of the list [15] (95.10) of
representations shows that C is locally isomorphic to an orthogonal group. Because
each action of SO5 R on a compact projective plane is trivial ([15] (55.40)), the group
C is simply connected and then C has a subgroup Y G Spin7 R. The central involution a A Y cannot be planar (or else Y would induce a group SO7 R on the fixed plane
Fa ). Hence a is a reflection with axis W and some center c. Because dim D c c 22, we
have dim c D d 10 and, therefore, dim a D a d 10. It is well-known that a D a is contained in the group T of translations with axis W and that a inverts each translation
in T. Consequently, Y acts faithfully on each invariant subgroup of T. There is only
one irreducible representation of Y in dimension c16, viz. the natural one on R8 . It
S156
Helmut Salzmann
follows that T G R16 is transitive and that dim TY ¼ 37. Finally, [4] Satz 3.6 or [15]
(81.17) shows that P G O.
9) Consider now the second case mentioned at the end of 7). By [15] (96.19), transitivity of ‘ on Ynf1g implies that a maximal compact subgroup F of ‘ is transitive
on the 7-sphere S consisting of all rays in Y. We know that SU3 C G L < F and that
dim F < dim ‘ c dim L þ t ¼ 16. From [15] (96.20–22) we can conclude that the
commutator group F 0 is isomorphic to SU4 C. Let o denote the central involution in
F 0 and note that F 0 =hoi G SO6 R. As in step 8), it follows that o is a reflection with
axis W , that the translation group T has dimension at least 10, and that T is the sum
of two 8-dimensional irreducible submodules; moreover, dim D ¼ dim ‘ þ dim T ¼
32, and the theorem is proved in the case t c 8.
10) For t > 8, the vector group Y contains a minimal normal subgroup H G Rs of
the connected component G of D av . Mutatis mutandis, the arguments in steps 3)–9)
can be applied to G and H instead of D and Y. Using the same notation as before, we
have
24 c dim G c dim a G þ dim %‘ þ dim L c 8 þ s þ dim L:
Hence (j) gives s d 2, moreover, L G G2 or s d 8.
11) Suppose that s < 8. As in step 4), it follows that ½L; H ¼ 1. Choose a point c
in the 2-dimensional subplane E with c A avnfa; vg. Then dim c H c 1 and Hc V L has
positive dimension, but L is simple. Therefore, s d 8. If s ¼ 8, the Theorem is true by
the arguments 7)–9).
12) To finish the proof, let s > 8 and consider the smallest closed subplane H
containing a H and u; v; w. If k is the dimension of a line of H, then k j 8. Note
that a H J av and that Ha induces the identity on H. It follows that dim Ha > 0,
hence H 0 P and k c 4. Since H has no compact subgroups other than 1, the sti¤ness theorem (j) shows that dim Ha < 8, moreover, dim Ha > 3 implies k c 2.
Only the possibility k ¼ 2 remains. By [15] (55.4), each closed subplane of H is
connected, and H‘ ¼ H because H is normal in G. There are points b; c A av V H
such that ‘b; c fixes H pointwise. On the other hand, dim ‘b; c d 12. This contradicts
the Corollary.
r
References
[1] R. Bödi, On the dimensions of automorphism groups of eight-dimensional ternary fields.
II. Geom. Dedicata 53 (1994), 201–216. MR 96c:51028 Zbl 0829.51007
[2] H. Freudenthal, H. de Vries, Linear Lie groups. Academic Press 1969. MR 41 #5546
Zbl 0377.22001
[3] T. Grundhöfer, H. Salzmann, Locally compact double loops and ternary fields. In:
Quasigroups and loops: theory and applications, 313–355, Heldermann, Berlin 1990.
MR 93g:20133 Zbl 0749.51016
[4] H. Hähl, Lokalkompakte zusammenhängende Translationsebenen mit großen Sphärenbahnen auf der Translationsachse. Resultate Math. 2 (1979), 62–87. MR 82a:51010
Zbl 0437.51011
[5] H. Hähl, SU4 ðCÞ als Kollineationsgruppe in sechzehndimensionalen lokalkompakten
Translationsebenen. Geom. Dedicata 23 (1987), 319–345. MR 88j:51023 Zbl 0622.51008
16-dimensional compact projective planes with 3 fixed points
S157
[6] H. Hähl, Sixteen-dimensional locally compact translation planes admitting SU4 C SU2 C
or SU4 C SL 2 R as a group of collineations. Abh. Math. Sem. Univ. Hamburg 70 (2000),
137–163. MR 2003b:51023 Zbl 0992.51007
[7] H. Hähl, Sixteen-dimensional locally compact translation planes with large automorphism groups having no fixed points. Geom. Dedicata 83 (2000), 105–117.
MR 2001h:51021 Zbl 0973.51011
[8] H. Hähl, H. Salzmann, 16-dimensional compact projective planes with a large group of
automorphisms fixing two points. In preparation.
[9] H. Löwe, 16-dimensional locally compact, connected translation planes admitting SL 2 H
as a group of collineations. To appear in Pacific J. Math. 209 (2003), 325–337.
[10] B. Priwitzer, Large semisimple groups on 16-dimensional compact projective planes are
almost simple. Arch. Math. (Basel ) 68 (1997), 430–440. MR 98e:51020 Zbl 0877.51014
[11] B. Priwitzer, Large almost simple groups acting on 16-dimensional compact projective
planes. Monatsh. Math. 127 (1999), 67–82. MR 2000d:51020 Zbl 0929.51010
[12] B. Priwitzer, H. Salzmann, Large automorphism groups of 16-dimensional planes are Lie
groups. J. Lie Theory 8 (1998), 83–93. MR 99f:51027 Zbl 0902.51012
[13] H. Salzmann, Characterization of 16-dimensional Hughes planes. Arch. Math. (Basel ) 71
(1998), 249–256. MR 99i:51013 Zbl 0926.51016
[14] H. Salzmann, Near-homogeneous 16-dimensional planes. Adv. Geom. 1 (2001), 145–155.
MR 2002h:51009 Zbl 1002.51011
[15] H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact
projective planes. de Gruyter 1995. MR 97b:51009 Zbl 0851.51003
Received 28 August, 2002
H. Salzmann, Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10,
72076 Tübingen, Germany
Email: helmut.salzmann@uni-tuebingen.de