Actions of almost simple groups on compact eight-dimensional projective planes are classical, almost

Stroppel, Markus

doi:10.1007/bf01265632

Cited by 6 publications

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“…9]. In [29] the results of the present paper are used in order to determine "almost all" projective planes. Proof, Let E be a subgroup of A that is isomorphic to SU2(C,0 ).…”

Section: The Symplectic Groupmentioning

confidence: 98%

The skew-hyperbolic motion group of the quaternion plane

Stroppel

1997

Monatshefte f�r Mathematik

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Abstract. Up to conjugation, there exist three different polarities of the projective plane P2 90 over Hamilton's quaternions H. The skew hyperbolic motion group of P2 H is introduced as the centralizer of a polarity "of the third kind". According to a result of R. L6wen, the qnaternion plane is characterized among the eight-dimensional stable planes by the fact that it admits an effective action of the centralizer of a polarity of the first or second kind (i.e., the elliptic or the hyperbolic motion group). In the present paper, we prove the analogous result for the skew hyperbolic case. Polarities of the Quaternion PlaneThis section collects basic information about polarities of the projective plane over Hamilton's quaternions. It seems that most of the contents of this section is folklore. As I could not find adequate references, I combine the introduction of the elliptic, hyperbolic and skew hyperbolic polarities with a proof that they represent the conjugacy classes of polarities of the quaternion plane.A polarity of a projective plane (~, A ~ is an involutory correlation, i.e., a mapping ~:: ~ • Ar -> A ~ ~ ~ such that ~2 is the identity mapping, and that K interchanges points and lines, but preserves incidence.In this first section, we are going to classify the polarities of the projective plane p2 H ~ (ul( ~ 3), u2(• 3)) over Hamilton's quaternions ~. Here u~(H 3) denotes the set of all n-dimensional vector subspaces of the left vector space ~ 3 consisting of rows x= (xl,x2,x3).If x= (xl,x2,x3)~(O,O,O), then the one-dimensional subspace spanned by x will be denoted by [x] = Ix1, x2, x3]. is called conjugation, it is an anti-automorphism of H. The eigenspaces ofp are ~ and Pull = ~i + Ej + Rk.1991 Mathematics Subject Classification: 51H20, 51A10.

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“…9]. In [29] the results of the present paper are used in order to determine "almost all" projective planes. Proof, Let E be a subgroup of A that is isomorphic to SU2(C,0 ).…”

Section: The Symplectic Groupmentioning

confidence: 98%

The skew-hyperbolic motion group of the quaternion plane

Stroppel

1997

Monatshefte f�r Mathematik

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“…Known results for 8-dimensional planes (e.g. Stroppel [16]) will then lead to a contradiction in each case. Interestingly, the arguments for di¨erent groups turn out to be rather di¨erent.…”

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“…Let q R t be a minimal normal subgroup of h. First it will be shown that t b 8. Then each point z e W is the center of some one-parameter subgroup of and the centralizer Cs coincides with T. Assuming that dim T` 16, it follows that the solvable radical h p has dimension at most 16, and a Levi complement g of (a maximal semi-simple subgroup of h) satis®es dim g 19. By minimality of , the group haT is an irreducible subgroup of GL t R. Either g acts irreducibly on , or splits into a direct product of two isomorphic minimal ginvariant subgroups, and the actions of g on the two factors are equivalent, compare [15, 95.6 (b)].…”

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“…Hence dim q 22. In each case, q H has a unique central involution s. According to Stroppel [16] or [15, 84.19], the group q H ahsi cannot act on an 8-dimensional plane, and s is not planar. Because s inverts the elements of , it follows that s is a re¯ection with axis W and center a.…”

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confidence: 99%

“…also [15, 71.10]. (16) Assume that g is almost simple and irreducible on R t , where still t 14. According to the List, g is a group of type C 3 , in fact, g is isomorphic to a motion group PU 3 HY r of the quaternion plane, or g is covered by the symplectic group Sp 6 R.…”

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Near-homogeneous 16-dimensional planes

2001

Advg

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A projective plane P PY L with point set P and line set L is a (compact) topological plane if P and L are (compact) topological spaces and the geometric operations of joining two distinct points and intersecting two distinct lines are continuous.The classical examples are the Desarguesian planes over the real or complex numbers or the quaternions and the Moufang plane over the octonions, each taken with its natural topology on P and on L. These planes are also connected. Within the vast class of compact connected topological projective planes they are distinguished by their high degree of homogeneity: their automorphism groups are transitive on quadrangles. A detailed discussion of the classical planes can be found in Chapter I of the book Compact Projective Planes [15].Each compact projective plane P with a point space P of positive (covering) dimension dim P`y has lines which are homotopy equivalent to an l-sphere S l , where l j 8 and dim P 2l, see [15, 54.11]. In all known examples, the lines are actually homeomorphic to S l . The automorphism group Aut P consists of all continuous collineations of PY L. Taken with the compact-open topology (the topology of uniform convergence), is a locally compact transformation group of P of ®nite dimension ([15, 83.2]). Let h denote a connected closed subgroup of . If dim h b 5l, then P is a classical plane and is a simple Lie group, see Salzmann [8], Theorem II,and [15, 87.7] for the cases l 4. Suppose now that 3l dim h 5l. Under suitable additional assumptions on the structure of h and its action on P, the classi®cation problem requires to determine all possible pairs PY h. The cases l j 4 are understood fairly well. In particular, the classi®cation has been completed in the following cases:

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Compact 16-dimensional projective planes

Salzmann¹

1999

Results. Math.

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This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in the book (which was published 20 years ago) is a complete description of all planes admitting an automorphism group of dimension at least 40. Newer results on 8-dimensional planes have been collected in [52]. Here, we present a classification of 16-dimensional planes with a group of dimension ≥35, provided the group does not fix exactly one flag, and we prove several further theorems, among them criteria for a connected group of automorphisms to be a Lie group. My sincere thanks are due to Hermann Hähl for many fruitful discussions. closure of an affine plane coordinatized by a mutation O (t) = (O, +, •) of the octonions, where c•z = t·cz + (1−t)·zc with t > 1 2 ; either t = 1, O (t) = O, and P ∼ = O , or dim Σ = 40 and Σ fixes the line at infinity and some point on this line ([54] 87.7).The ultimate goal is to describe all pairs (P, ∆), where ∆ is a connected closed subgroup of Aut P and dim ∆ ≥ b for a suitable bound b in the range 27 ≤ b < 40. This bound varies with the structure of ∆ and the configuration F ∆ of the fixed elements (points and lines) of ∆. In some cases, only ∆ can be determined, but there seems to be no way to find an explicit description of all the corresponding planes.

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Actions of almost simple groups on compact eight-dimensional projective planes are classical, almost

Abstract: Abstract. We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quartemion plane) among the 8-dimensional compact projective planes.Mathematics Subject Classifications (1991): 51H20, 51A 10.

Cited by 6 publications

References 29 publications

The skew-hyperbolic motion group of the quaternion plane

The skew-hyperbolic motion group of the quaternion plane

Near-homogeneous 16-dimensional planes

Compact 16-dimensional projective planes

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