(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
LetP¼ ðP;LÞbe a topological projective plane with a compact point setPof finite (covering) dimension d ¼dimP>0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line LAL is homotopy equivalent to a sphereSlwithlj8, andd ¼2l, see [15] (54.11). In all known exam- ples, Lis in fact homeomorphic to Sl. Taken with the compact-open topology, the automorphism groupS¼AutP(of all continuous collineations) is a locally compact transformation group ofPwith a countable basis, the dimension dimSis finite [15]
(44.3 and 83.2).
The classical examples are the planesPKover the three locally compact, connected fields K with l¼dimK and the 16-dimensional Moufang plane O¼PO over the octonion algebra O. If P is a classical plane, then AutP is an almost simple Lie group of dimensionCl, whereC1¼8,C2¼16,C4¼35, andC8¼78.
In all other cases, dimSc12Clþ1c5l. Planes with a group of dimension su‰- ciently close to12Clcan be described explicitly. More precisely,
the classification program seeks to determine all pairsðP;DÞ,whereDis a connected closed subgroup ofAutPand blcdimDc5lfor a suitable bound bld4l1.
This has been accomplished for lc2 and also for b4¼17. Here, the casel¼8 will be studied; the value of bl varies with the configuration of the fixed elements ofD.
Most theorems that have been obtained so far require additional assumptions on the structure ofD. If dimDd27, thenDis always a Lie group [12].
By the structure theory of Lie groups, there are 3 possibilities: (i)Dis semi-simple, or (ii) D contains a central torus subgroup, or (iii) Dhas a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well:
(a) If D is semi-simple and dimD>28, then DGSL3H and P is a Hughes plane (as described in [15] §86),orDGSpin9ðR;rÞwith rc1,orPGO, see [10], [11].
(b) IfDcontains a central torus,and ifdimD>30,thenD0GSL3H, see [13].
A groupD of type (iii) fixes a point or a line, cf. [3] (XI.10.19). Hence (a) and (b) imply
(c) If dimD>30 and D has no fixed element, then P is a Hughes plane or PGO.
The case thatDfixes exactly one element has been treated in [14]:
(d) IfdimDd35and ifDfixes one line and no point,thenPis a translation plane.
All such planes have been determined in [6], [7], [9]. EitherPGOor dimD¼35.
Little progress has been made in the cases where D fixes exactly two elements, necessarily a point and a line. If dimDd40, then P and its dual are translation planes [15] (87.7). Alltranslation planeswith dimDd38 are described in [15] (82.28).
(e) If dimDd34and D fixes exactly 2 points and only one line, then D contains a translation group of dimension at least15.
(f ) IfdimDd33andDfixes2points and2lines,thenDcontains a translation group TGR8 and a compact subgroupFGSpin8R.
A method to construct all planes with exactly 2 fixed points have been given in [8].
A smaller dimension ofDsu‰ces ifDfixes more than two points (the last case to be considered):
Theorem.IfdimDd32andDhas(at least) 3fixed points,thenDcontains a transitive translation groupT.EitherdimD¼32and a maximal semi-simple subgroupCofDis isomorphic toSU4C,ordimDd37andPGO.
Translation planes with a group CGSU4C have already been studied in [5].
Examples of proper translation planes such that TChas a fixed point setSAS2are given in [6].
According to the sti¤nessresult [15] (83.23), the stabilizerLof a non-degenerate quadrangle satisfies dimLc14. The proof of the theorem depends decisively on Bo¨di’s improvement [1] of the sti¤ness theorem:
(j) If the fixed elements of the connected Lie group L form a connected subplane E,thenLis isomorphic to the14-dimensional compact groupG2 or its subgroup SU3CordimL<8.IfEis a Baer subplane(dimE¼8),thenLis a subgroup of SU2C.
Corollary.FromdimL>8it follows thatdimE¼2.
Proof.Assume that dimE¼4. IfLis any line of Eand ifcALnE, then dimLc>0 and the fixed elements of Lc form a Baer subplanehE;ci. Hence dimLcc3 and dimLc11. An alternative proof is given by [15] (96.35). r Proof of the Theorem. 1) For any closed subgroup GcD and any point x the dimension formula dimG¼dimGxþdimxGholds, 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‘c30. Hence
all fixed points of D are incident with the same line W. There are at least 3 fixed pointsu;v;wAW and the sti¤ness theorem implies dimDc38.
3) Because of results (a) and (b), the group D has a minimal normal subgroup YGRt. Choose aBW and%APcYsuch that PGR and a%0a. Since D acts linearly on Y, the centralizer Cs% is also the centralizer of P, and the dimension formula gives dim CsPd32t. The connected componentLofDaVCsPfixes the orbitaPpointwise, and the fixed elements ofLform a connected subplaneE, see [15]
(42.1). By (j) we have dimDatcdimLc14 and td2; moreover, dimL¼14 or dimLc8.
4) Assume first that t<8. Then LGG2 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 G2onR16has a dimension divisible by 7, see [15] (95.10). Since PL¼P, it follows from tc7 that the commutator
½L;Yis trivial.
5) The last statement implies that the orbit aY is contained in E. Because Yis commutative,Ya fixes each point of aY. Hence Ya acts trivially on the subplane E generated byaPandu;v;w, and the connected component ofYa is contained inL, butLis simple andLVY¼1. Therefore, dimYa¼0 and dimaY¼t¼2.
6) Denote the connected component ofDaby‘. From steps 3) and 5) it follows that dim‘¼16. Consequently, ‘ has a 2-dimensional radical P¼ ffiffiffiffi
p‘
, and ½L;P ¼1.
HenceEP¼E. Ifcis a point ofEandcAawnfa;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 withEby [15] (32.7); thus the connected component of Pcwould belong to the simple groupL. This contradiction shows thattd8.
7) Ift¼8, then 16cdim‘¼dim%‘þdimLctþ14¼22 and dimLd8. Con- sider the smallest closed subplane F containing aY and u;v;w, and assume that P0F¼F‘. Then‘induces onFa group‘=K of dimensionc7, see [15] (83.17).
Hence dim Kd9 and K contains G2. The Corollary implies that dimF¼2 and then dim‘=Kc1 and dim K>14. This contradiction shows F¼P and Ya¼1 (becauseYa fixesFpointwise). By (j) there are two possibilities: eitherLGG2 for some%AY, orLGSU3Cfor each choice of%, and‘acts transitively onYnf1gby [15] (96.11). These cases will be treated separately.
8) Suppose thatLGG2 and thatLis contained in the maximal semi-simple sub- groupCofD. By minimality ofYand [15] (95.6b), the groupCacts irreducibly onY 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 thatCis locally isomorphic to an orthogonal group. Because each action of SO5Ron a compact projective plane is trivial ([15] (55.40)), the group Cis simply connected and thenChas a subgroup YGSpin7R. The central involu- tionaAY cannot be planar (or else Y would induce a group SO7Ron the fixed plane Fa). Henceais a reflection with axisW and some centerc. Because dimDcc22, we have dimcDd10 and, therefore, dimaDad10. It is well-known that aDa is con- tained in the group T of translations with axisW and thatainverts each translation in T. Consequently, Y acts faithfully on each invariant subgroup of T. There is only one irreducible representation of Y in dimensionc16, viz. the natural one onR8. It
follows that TGR16 is transitive and that dim TY¼37. Finally, [4] Satz 3.6 or [15]
(81.17) shows thatPGO.
9) Consider now the second case mentioned at the end of 7). By [15] (96.19), tran- sitivity of‘onYnf1gimplies that a maximal compact subgroupFof‘is transitive on the 7-sphereSconsisting of all rays inY. We know that SU3CGL<Fand that dimF<dim‘cdimLþt¼16. From [15] (96.20–22) we can conclude that the commutator groupF0is isomorphic to SU4C. Letodenote the central involution in F0and note thatF0=hoiGSO6R. As in step 8), it follows thatois a reflection with axisW, that the translation group T has dimension at least 10, and that T is the sum of two 8-dimensional irreducible submodules; moreover, dimD¼dim‘þdim T¼ 32, and the theorem is proved in the casetc8.
10) Fort>8, the vector groupYcontains a minimal normal subgroup HGRsof the connected componentGofDav. Mutatis mutandis, the arguments in steps 3)–9) can be applied toGand H instead ofDandY. Using the same notation as before, we have
24cdimGcdimaGþdim%‘þdimLc8þsþdimL:
Hence (j) givessd2, moreover,LGG2orsd8.
11) Suppose thats<8. As in step 4), it follows that½L;H ¼1. Choose a pointc in the 2-dimensional subplaneEwithcAavnfa;vg. Then dimcHc1 and HcVLhas positive dimension, butLis simple. Therefore,sd8. Ifs¼8, 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 aH and u;v;w. If k is the dimension of a line of H, then kj8. Note that aHJav and that Ha induces the identity on H. It follows that dim Ha>0, henceH0Pandkc4. Since H has no compact subgroups other than1, the sti¤- ness theorem (j) shows that dim Ha<8, moreover, dim Ha>3 implies kc2.
Only the possibility k¼2 remains. By [15] (55.4), each closed subplane of H is connected, andH‘¼H because H is normal in G. There are pointsb;cAavVH such that‘b;cfixesHpointwise. On the other hand, dim‘b;cd12. This contradicts
the Corollary. r
References
[1] R. Bo¨di, On the dimensions of automorphism groups of eight-dimensional ternary fields.
II.Geom. Dedicata53(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. Grundho¨fer, 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. Ha¨hl, Lokalkompakte zusammenha¨ngende Translationsebenen mit großen Spha¨r- enbahnen auf der Translationsachse.Resultate Math.2(1979), 62–87. MR 82a:51010 Zbl 0437.51011
[5] H. Ha¨hl, SU4ðCÞ als Kollineationsgruppe in sechzehndimensionalen lokalkompakten Translationsebenen.Geom. Dedicata23(1987), 319–345. MR 88j:51023 Zbl 0622.51008
[6] H. Ha¨hl, Sixteen-dimensional locally compact translation planes admitting SU4CSU2C or SU4CSL2Ras a group of collineations.Abh. Math. Sem. Univ. Hamburg70(2000), 137–163. MR 2003b:51023 Zbl 0992.51007
[7] H. Ha¨hl, Sixteen-dimensional locally compact translation planes with large automor- phism groups having no fixed points.Geom. Dedicata83(2000), 105–117.
MR 2001h:51021 Zbl 0973.51011
[8] H. Ha¨hl, H. Salzmann, 16-dimensional compact projective planes with a large group of automorphisms fixing two points. In preparation.
[9] H. Lo¨we, 16-dimensional locally compact, connected translation planes admitting SL2H as a group of collineations. To appear inPacific 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 Theory8(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. Grundho¨fer, H. Ha¨hl, R. Lo¨wen, M. Stroppel, Compact projective planes. de Gruyter 1995. MR 97b:51009 Zbl 0851.51003
Received 28 August, 2002
H. Salzmann, Mathematisches Institut der Universita¨t Tu¨bingen, Auf der Morgenstelle 10, 72076 Tu¨bingen, Germany
Email: [email protected]
