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  • 1
    Electronic Resource
    Electronic Resource
    An Essay on the Ree Octagons (1995)
    Springer
    Journal of algebraic combinatorics 4 (1995), S. 145-164 
    Details
    ISSN: 1572-9192
    Keywords: Tits building ; generalized octagon ; group of projectivities ; coordinatization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We coordinatize the Moufang generalized octagons arising from the Ree groups of type 2 F 4. In this way, we obtain a very concrete and explicit description of these octagons. We use this to prove some results on suboctagons, generalized homologies, Suzuki-Tits ovoids and groups of projectivities of the Ree octagons. All our results hold for arbitrary Ree octagons, finite or not.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022481514132
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  • 2
    Electronic Resource
    Electronic Resource
    Translation Ovoids of Generalized Quadrangles and Hexagnos (1998)
    Springer
    Geometriae dedicata 72 (1998), S. 19-62 
    Details
    ISSN: 1572-9168
    Keywords: ovoids ; spreads ; classical generalized polygons.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=32h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any ≡ 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005011600115
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  • 3
    Electronic Resource
    Electronic Resource
    Characterizations by Automorphism Groups of some Rank 3 Buildings – I. Some Properties of Half Strongly-Transitive Triangle Buildings (1998)
    Springer
    Geometriae dedicata 73 (1998), S. 119-142 
    Details
    ISSN: 1572-9168
    Keywords: affine building ; triangle building ; Bruhat–Tits building ; Hjelmslev plane ; stgrongly transitive group.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building Δ forces Δ to be one of the examples arising from PSL3(K) for a locally finite local skewfield K. Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005009315851
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  • 4
    Electronic Resource
    Electronic Resource
    Sub-weakly Embedded Singular and Degenerate Polar Spaces (1997)
    Springer
    Geometriae dedicata 65 (1997), S. 291-298 
    Details
    ISSN: 1572-9168
    Keywords: polar space ; embedding.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG $$(d,\mathbb{K}) $$ , $$\mathbb{K} $$ a commutative field, is the projection of a full embedding in some subspace PG $$\left( \overline {d}, \mathbb{F} \right) $$ of PG $$\left( \overline {d}, \mathbb{K} \right) $$ , where PG $$\left( \overline {d}, \mathbb{K} \right) $$ contains PG $$(d,\mathbb{K}) $$ and $$\mathbb{F} $$ is a subfield of $$\mathbb{K} $$ . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space Γ contains exactly two points of Γ. This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field $$\mathbb{F}\prime $$ , where in the characteristic 2 case $$\mathbb{F}\prime $$ is perfect if the polar space is symplectic and the degree of the embedding is 2.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1017997605039
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