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Proceedings of the London Mathematical Society 2006 93(3):666-692; doi:10.1017/S0024611506015851
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© London Mathematical Society

Intransitive Geometries

Ralf Gramlich and Hendrik van Maldeghem

FB Mathematik/AG 5, TU Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt, Germany gramlich{at}mathematik.tu-darmstadt.de
Pure Mathematics and Computer Algebra, Ghent University Krijgslaan 281, S22, 9000 Gent, Belgium hvm{at}cage.ugent.be

Received 20 October 2004. Revision received 9 November 2005.

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples. 2000 Mathematics Subject Classification 20E06, 05E20, 05E25, 51A05.


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