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, Curtis–Phan–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 ( 2 n + 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 ( 2 n + 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.