Even-dimensional orthogonal groups as amalgams of unitary groups

Abstract

This article is part of the program described in [C.We provide presentations of even-dimensional orthogonal groups, a characterization of even-dimensional orthogonal groups by certain amalgams of subgroups isomorphic to SU(2, q 2 ), SU(3, q 2 ), and SU(2, q 2 ) × SU(2, q 2 ) over any finite field, and a classification of related amalgams. The results are obtained by diagram geometry, geometric covering theory, and analysis of amalgams.

On flips of unitary buildings I: Classification of flips

“…On the other hand, since both B 1 and B 2 are β hyperbolic bases, we see that B 1 = B 2 , which combined with (7) forces N σ (det(T )) = 1, a contradiction. Hence M and M ′ have the same β ϕ type.…”

“…These so-called "Phan-type" theorems have been studied in a number of papers (e.g. [4], [3], [7]) initially in order to aid the Gorenstein-Lyons-Solomon revision of the proof of the Classification of Finite Simple Groups. Roughtly speaking, these "Phantype" theorems allow for the recognition of a group based on amalgams of subgroups that are produced by the group acting on a geometry.…”

Opposition in triality

“…By [10,Lemma 5.4], it is possible to show that G is simply connected by studying the point-line geometry of G. In order to establish the simple connectedness of G we have to prove that every closed path (also called circuit) in the collinearity graph G is a sum of triangles (triples of collinear points) contained in objects of type 3 or 4. (If a triple is contained in an object of type 2, it is contained in an object of type 3 or 4.)…”

Developments in finite Phan theory

“…Consider the situation as in Example 4a, but over the field F q 2 , and let G = Ω + 2n (q 2 ). For sake of simplicity of the exposition we assume here that q is odd, although in [71] also the case of even characteristic is dealt with. The Phan involution τ can again be defined as the composition of the linear transformation given by the Gram matrix of the bilinear form (·, ·) with respect to a hyperbolic basis and coordinate-wise application of the involutory field automorphism.…”

“…This τ produces a flipflop geometry on which G τ ∼ = Ω ± 2n (q) acts flag-transitively, cf. [71,Proposition 3.10]. The geometry G τ can be described as follows.…”