Large automorphism groups of 8-dimensional projective planes are Lie groups

2015

J. Geom.

The automorphism group Σ of a compact topological projective plane with a 16-dimensional point space is a locally compact group. If the dimension of Σ is at least 29, then Σ is known to be a Lie group. For the connected component ∆ of Σ the condition dim ∆ ≥ 27 suffices. Depending on the structure of ∆ and the configuration of the fixed elements of ∆ sharper bounds are obtained here. Example: If ∆ is semi-simple and fixes exactly one line and possibly several points on this line, then ∆ is a Lie group if dim ∆ ≥ 11. Any semi-simple group ∆ which satisfies dim ∆ ≥ 25 is a Lie group.

Section: No Fixed Elementsmentioning

confidence: 99%

“…(b) Suppose now that x ζ = x for some x / ∈ W and some ζ ∈ N\1l. By assumption, x ∆ is not contained in a line and hence generates a ∆- [9]), and we may assume that 1l = N ≤ K. If even dim ∆ * > 16, then [16] 1.10 shows that D is the classical quaternion plane, because ∆ * does not fix a flag.…”

Section: Exactly One Fixed Elementmentioning

confidence: 99%

Semi-simple groups of compact 16-dimensional planes

2015

J. Geom.

“…In many cases, Σ is known to be even a Lie group, cf. [11] (87.1) and [6]. Here, we deal exclusively with 16-dimensional planes; then the full automorphism group Σ is a Lie group whenever dim Σ ≥ 29 , see [9].…”

mentioning

confidence: 99%

16-dimensional compact projective planes with a collineation group of dimension ≥ 35

2008

Arch. Math.

A compact projective plane is isomorphic to the classical Moufang plane if it admits an automorphism group of dimension at least 38 which fixes a non-incident point-line pair.Let P = (P, L) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0 . A systematic treatment of such planes can be found in the book Compact Projective Planes [11]. Each line L ∈ L is homotopy equivalent to a sphere S with | 8 , and d = 2 , see [11] (54.11). In all known examples, L is in fact homeomorphic to S . Taken with the compact-open topology, the automorphism group Σ = Aut P (of all continuous collineations) is a locally compact transformation group of P with a countable basis, the dimension dim Σ is finite [11] (44.3 and 83.2). In many cases, Σ is known to be even a Lie group, cf. [11] (87.1) and [6]. Here, we deal exclusively with 16-dimensional planes; then the full automorphism group Σ is a Lie group whenever dim Σ ≥ 29 , see [9].The last theorem in the book [11] states that a plane with dim Σ ≥ 40 can be coordinatized by a "mutation" of the octonions with a multiplication c • z = (1−τ )cz +τ zc, where 1 2 = τ ∈ R . In the meantime, considerable progress has been made towards a classification of compact planes admitting a connected group ∆ of automorphisms with dim ∆ ≥ 35 . All such planes have been determined in which ∆ does not fix exactly one point and one line, see [2], [3] and the references given there. Here we will deal with the case that ∆ fixes exactly two non-incident elements. The following is already known and will be used:

“…Most theorems that have been obtained so far require additional assumptions on the structure and/or the action of ∆. If dim ∆ 27 , and in the relevant dimension range in particular, ∆ is always a Lie group [15]. By the structure theory of Lie groups, ∆ is semi-simple, or ∆ contains a central torus subgroup, or ∆ has a minimal normal vector subgroup, cf.…”

mentioning

confidence: 99%

16-dimensional compact projective planes with a large group fixing two points and two lines

Hähl

2005

Arch. Math.

H e r r n P r o f e s s o r O t t o H . K e g e l z u m 7 0. G e b u r t s t a g Abstract. We determine all planes having the properties of the title with a group of dimension at least 33.Let P = (P , L) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0 . A systematic treatment of such planes can be found in the book Compact Projective Planes [21]. Each line L ∈ L is homotopy equivalent to a sphere S with | 8 , and d = 2 , see [21, (54.11)]. In all known examples, L is in fact homeomorphic ( ≈ ) to S . Taken with the compact-open topology, the automorphism group Σ = Aut P (of all continuous collineations) is a locally compact transformation group of P with a countable basis [21, (44.3)]. The covering dimension dim Σ is an important parameter for characterizations of such planes. (For readers which are more familiar with the inductive dimension, we remark that for a locally compact group Λ, the inductive dimension ind Λ coincides with dim Λ and with the dimension of the connected component Λ 1 , cf. [21, (93.5, 6)].The classical examples are the planes P K over the 3 locally compact, connected fields K with = dim K and the 16 -dimensional Moufang plane O = P O over the octonion algebra O . If P is a classical plane, then Aut P is an almost simple Lie group of dimension C , where C 1 = 8, C 2 = 16, C 4 = 35 , and C 8 = 78 .In all other cases, dim Σ 1 2 C + 1 5 . Planes with small groups abound, those with a group of dimension sufficiently close to 1 2 C can be described explicitly. More precisely, the classification program seeks to determine all pairs (P, ∆) , where ∆ is a connected closed subgroup of Aut P and b dim ∆ 5 for a suitable bound b 4 − 1 . This has been accomplished for 2 and also for b 4 = 17 . Results in the case = 8 are as yet less satisfactory, and it is this case that will be considered here. (2000): 51H10, 51A35, 12K99. Mathematics Subject Classification