TY - JOUR
T1 - Odd-dimensional orthogonal groups as amalgams of unitary groups. I. General simple connectedness
AU - Gramlich, Ralf
AU - Hoffman, Corneliu
AU - Shpectorov, Sergey
AU - Bennett, C
PY - 2007/6/1
Y1 - 2007/6/1
N2 - 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. Sax1 (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-q2. 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 is an element of {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. (C) 2007 Elsevier Inc. All rights reserved.
AB - 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. Sax1 (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-q2. 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 is an element of {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. (C) 2007 Elsevier Inc. All rights reserved.
U2 - 10.1016/j.jalgebra.2007.02.011
DO - 10.1016/j.jalgebra.2007.02.011
M3 - Article
VL - 312
SP - 426
EP - 444
JO - Journal of Algebra
JF - Journal of Algebra
IS - 1
ER -