Compact Projective Planes: \bf Recent developments

Recent developments

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Click here to go directly to remarks on Chapter 1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, or the Bibliography.


Regarding Chapter 1:

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Regarding Chapter 2:

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Regarding Chapter 3:

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Regarding Chapter 4:

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Regarding Chapter 5:

[CPP: 55.38] can be improved: Two closed Baer subplanes always have a point and a line in common, not only if they are the fixed point sets of involutions. See R. Löwen [1998b].

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Regarding Chapter 6:

[CPP: 65.2] can be improved: it suffices to assume that dimS Cl-2. The following additional argument is then required in step 1) of the proof:
If B is contained in a line L through x, then Sx=Sx,L and, for a suitable s S, the stabilizer Sx,xs fixes a triangle, but dimSx,xs Cl-2-4l is too large.

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Regarding Chapter 7:

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Regarding Chapter 8:

8.a.
The following results improve [CPP: 87.1]:

If P is a 16-dimensional plane, and if dimD 27, then D is a Lie group.
See B. Priwitzer - H. Salzmann [1998].

Irrespective of connectedness, the automorphism groups of a 16-dimensional projective plane is a Lie group, if it has dimension at least 29; see H. Salzmann [1999a].

8.b.
Regarding [CPP: 87.2] we remark that the same results have been obtained in B. Priwitzer [1997] and B. Priwitzer [1998] under the weaker hypothesis that dimD > 28 (instead of 30).

8.c.
[CPP: 87.3] is true for dimD 31 instead of 35. See H. Salzmann [1998].

8.d.
ad [CPP: 87.4]: the arguments on p.587 can be improved as follows:
Theorem. If D is not semi-simple, and if dimD 33, then, up to duality, D has a minimal normal subgroup Q @ Rt consisting of axial collineations with common axis W.
Either Q D[a,W] is a group of homologies and t=1, or Q is contained in the group T = D[W,W] of elations with axis W.

For full proofs see H. Salzmann [1999b].

8.e.
The results described in Section 87 have been extended, as follows.
Theorem. If dimD 35 and if D fixes exactly one line W and no point, then P is a translation plane.
Recently, Hähl and Löwe have determined explicitly all translation planes having a group D as in this theorem. In particular, their work implies the following:
Corollary.Under the assumptions of the Theorem, either P @ P2O, or dimD = 35 and the stabilizer of an affine point has an 18-dimensional semisimple commutator group U isomorphic to one of the groups SL2H·SU2C or SU4C·D with D=SU2C or D=SL2R.
If D fixes two distinct points, and if dimD = 39, then P is a translation plane over a perturbation of the octonions
[CPP: 82.5b]; see H. Salzmann [1999b].

8.f.
The following results are obtained in H. Salzmann [1999d]:
Lemma. Assume that dimD 29. If D fixes no point and no line, then P is classical, or D @ SL3(H) and P is a Hughes plane. If D fixes exactly one element, then D has a normal vector subgroup.
Theorem. If dimD 35 and if D fixes exactly one line W and no point, then P is a translation plane.
Corollary. If dimD 38 and if D fixes exactly one element (point or line), then P is isomorphic to the octonion plane.

8.g.
Moreover, the following is true (H. Salzmann [2000-2003]):

Theorem. If dimD 34 and if D fixes exactly 2 points u,v and the line uv, then the group T of translations with axis uv is at least 15-dimensional. Either D has a subgroup Y @ Spin7 R and dimD 36, or T is transitive, a maximal semi-simple subgroup of D is isomorphic to SU4 C @ Spin6R, and dimD = 34 .

Theorem. If D fixes exactly 2 points u,v and 2 lines W=uv and Y=av, then the translation group T=D[v,W] is transitive, the complement Da of T has a compact commutator group F @ Spin8 R, and dimD 36. If even dimD 38, then P is the classical Moufang plane .

Theorem. If dimD 32 and D has ( at least ) 3 fixed points, then D contains a transitive translation group T. Either dimD = 32 and a maximal semi-simple subgroup Y of D is isomorphic to SU4 C, or dimD 37 and P is the classical Moufang plane .

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Regarding Chapter 9:




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