Recent developments

Regarding Chapter 1:

Regarding Chapter 2:

Regarding Chapter 3:

Regarding Chapter 4:

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].

Regarding Chapter 6:

[CPP: 65.2] can be improved: it suffices to assume that dim Σ ≥ C_{ℓ} − 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 Σ_{x} = Σ_{x,L} and, for a suitable σ ∈ Σ, the stabilizer Σ_{x,x^{σ}} fixes a triangle, but dim Σ_{x,x^{σ}} ≥ C_{ℓ} − 2 − 4ℓ is too large.

Regarding Chapter 7:

Regarding Chapter 8:

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

If P is a 16-dimensional plane, and if dim Δ ≥ 27, then Δ 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 dim Δ > 28 (instead of 30).

8.c. [CPP: 87.3] is true for dim Δ ≥ 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 Δ is not semi-simple, and if dim Δ ≥ 33, then, up to duality, Δ has a minimal normal subgroup Θ ≅ R^{t} consisting of axial collineations with common axis W. Either Θ ≤ Δ_{[a,W]} is a group of homologies and t = 1, or Θ is contained in the group T = Δ_{[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 dim Δ ≥ 35 and if Δ 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 Δ as in this theorem. In particular, their work implies the following: Corollary.Under the assumptions of the Theorem, either P ≅ P_{2}O, or dim Δ = 35 and the stabilizer of an affine point has an 18-dimensional semisimple commutator group Υ isomorphic to one of the groups SL_{2}H·SU_{2}C or SU_{4}C·D with D = SU_{2}C or D = SL_{2}R. If Δ fixes two distinct points, and if dim Δ = 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 dim Δ ≥ 29. If Δ fixes no point and no line, then P is classical, or Δ′ ≅ SL_{3}(H) and P is a Hughes plane. If Δ fixes exactly one element, then Δ has a normal vector subgroup. Theorem. If dim Δ ≥ 35 and if Δ fixes exactly one line W and no point, then P is a translation plane. Corollary. If dim Δ ≥ 38 and if Δ 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 dim Δ ≥ 34 and if Δ 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 Δ has a subgroup Y ≅ Spin_{7}R and dim Δ ≥ 36, or T is transitive, a maximal semi-simple subgroup of Δ is isomorphic to SU_{4}C ≅ Spin_{6}R, and dim Δ = 34.

Theorem. If Δ fixes exactly 2 points u, v and 2 lines W = uv and Y = av, then the translation group T = Δ_{[v,W]} is transitive, the complement Δ_{a} of T has a compact commutator group Φ ≅ Spin_{8}R, and dim Δ ≥ 36. If even dim Δ ≥ 38, then P is the classical Moufang plane.

Theorem. If dim Δ ≥ 32 and Δ has (at least) 3 fixed points, then Δ contains a transitive translation group T. Either dim Δ = 32 and a maximal semi-simple subgroup Ψ of Δ is isomorphic to SU_{4}C, or dim Δ ≥ 37 and P is the classical Moufang plane.

Regarding Chapter 9: