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[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].
[CPP: 65.2] can be improved: it suffices to assume that
dimS ³ 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 Î S, the stabilizer Sx,xs fixes a
triangle, but dimSx,xs ³ Cl-2-4l is too large.
8.a.
The following results improve [CPP: 87.1]:
If P is a 16-dimensional plane, and if dimD ³ 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 ³ 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 ³ 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 ³ 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 ³ 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 ³ 35 and if D fixes
exactly one line W and no point,
then P is a translation plane.
Corollary. If dimD ³ 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 ³ 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 ³ 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 ³ 36. If even dimD ³ 38, then P is the classical Moufang plane .
Theorem. If dimD ³ 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 ³ 37 and P is the classical Moufang plane .