On the Foundations of Incidence Geometry

Buekenhout, Francis; Buset, Dominique

doi:10.1007/978-94-009-4017-8_7

Cited by 8 publications

(21 citation statements)

References 7 publications

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“…Finally, define the partial order by Γ j ϕ ≤ Γ k ψ if and only if −1 ≤ j ≤ k ≤ r and Γ j ϕ ∩ Γ k ψ = ∅. This construction identifies P as a particular kind of thin diagram geometry (see [4,5]). …”

Section: Abstract Regular Polytopesmentioning

confidence: 98%

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

2015

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regular polytopes Thin regular geometries Almost simple groups of PSL(2, q) type Our paper deals with the classification of abstract regular polytopes for almost simple groups with socle PSL(2, q). We consider all almost simple groups PSL(2, q) ≤ G ≤ PΓL(2, q) and determine the maximal rank of string C-group representations for G, i.e. the maximal rank of an abstract regular polytope with automorphism group G, as well as the existence of string C-group representations of lower ranks. Similar results have already been obtained by various authors in the cases G ∼ = PSL(2, q) and G ∼ = PGL(2, q) and they are summarized in this paper.

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Section: Abstract Regular Polytopesmentioning

confidence: 98%

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

2015

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“…All our geometries will have a string diagram. This means that these geometries have a diagram as in Buekenhout [Bu95] and this diagram will look like a string, or a simple path. For convenience of the reader we will give the following equivalent ad-hoc definition here.…”

Section: Geometriesmentioning

confidence: 97%

A quasi Curtis–Tits–Phan theorem for the symplectic group

Blok

Hoffman

2008

Journal of Algebra

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We obtain the symplectic group Sp(V ) as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let Sp(V ) act flag-transitively on the geometry of maximal rank subspaces of V . We show that this geometry and its rank 3 residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups.

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“…We will show that A and A are isomorphic by constructing three further isomorphisms, ϕ p , ϕ L and ϕ π satisfying Definition 6.1 (1) and (2).…”

Section: Amalgams Of Type A(e/f)mentioning

confidence: 99%

On a class of affine geometries

Hoffman¹,

Parker²,

Shpectorov³

2012

Advances in Geometry

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The far-away geometry Σ in a symplectic polar space is the subgeometry on the points non-collinear to a fixed point. In case of the polar space of rank three we define a class of subgeometries of Σ with the same points and lines as Σ and some of the planes of Σ removed. We show that such geometries are simply connected, unless Σ is defined over GF(2). As an application, for a particular such subgeometry Γ we exhibit a flag-transitive group G and, as a result, we get an amalgam presentation for this G. The smallest instance (over GF (3)) of G is related to the sporadic Thompson simple group.

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On the Foundations of Incidence Geometry

Cited by 8 publications

References 7 publications

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

Almost simple groups with socle PSL(2,q) acting on abstract regular polytopes

A quasi Curtis–Tits–Phan theorem for the symplectic group

On a class of affine geometries

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