Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

Abstract

We extend the Phan theory described in [C. Bennett, R. Gramlich, C. Hoffman, S. Shpectorov, CurtisPhan-Tits theory, in: A.A. Ivanov, M.W. Liebeck, J. Saxl (Eds.), Groups, Combinatorics, and Geometry, World Scientific, River Edge, 2003, pp. 13-29] to the last remaining infinite series of classical Chevalley groups over finite fields. Namely, we prove that the twin buildings for the group Spin(2n + 1, q 2 ), q odd, admit a unique unitary flip and that the corresponding flipflop geometry is simply connected for almost all finite fields F q 2 . Applying standard methods from amalgam theory, this results in a characterization of central quotients of the group Spin(2n + 1, q) by a Phan system of rank one and rank two subgroups. In the present first part of a series of two articles we present simple connectedness results for sufficiently large fields or sufficiently large rank. To be precise, the result stated in the present paper is proved for all cases but n = 3 and q ∈ {3, 5, 7, 9}, the remaining cases are dealt with in the sequel [R. Gramlich, M. Horn, W. Nickel, Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: Machine computations, submitted for publication] computationally.

Odd-dimensional orthogonal groups as amalgams of unitary groups.

“…Part 1 [Part1] of this work proves this theorem except for the cases n = 3 and 5 q 9. For n = q = 3 the geometry Γ is not simply connected but rather admits a 2187-fold universal cover.…”

On flips of unitary buildings I: Classification of flips

“…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.…”

On the Phan system of the Schur cover of SU(4, 32)

“…In Sect. 3, we analyze the amalgam of rank two parabolics for (n, q) = (3, 3) by applying methods from computer algebra, similar to those previously used by the author in [10,14] (for C n ) and [11] (for B n ; see also [4]). It is assumed throughout the paper that the reader is familiar with the concept of amalgams (refer to [17] for an introduction to the subject).…”

“…We now describe the exact structure of H . 4 The group H is isomorphic to a non-split central extension of SU(4, 3 2 ) by K, i.e. the following sequence is exact and non-split:…”