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 -