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Israel Journal of Mathematics

, Volume 109, Issue 1, pp 189–224 | Cite as

Simple groups of finite morley rank and Tits buildings

  • Linus Kramer
  • Katrin Tent
  • Hendrik Van Maldeghem
Article

Abstract

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon.

It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL 3(K),PSp 4(K) orG 2(K) for some algebraically closed fieldK.

Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show

Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field.

Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed.

We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

Keywords

Spherical Building Schubert Cell Coxeter Complex Morley Rank Root Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Magnes Press 1999

Authors and Affiliations

  • Linus Kramer
    • 1
  • Katrin Tent
    • 1
  • Hendrik Van Maldeghem
    • 2
  1. 1.Mathematisches Institut Universität Würzburg Am HublandWürzburgGermany
  2. 2.Department of Pure Mathematics and Computer AlgebraUniversity of Ghent Galglaan 2GhentBelgium

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