Simple groups of finite morley rank and Tits buildings
Copyright policy )Simple groups of finite morley rank and Tits buildings
The Morley rank of a structure is a model-theoretic dimension function which measures the complexity of the definable subsets of that structure. Finiteness of the Morley rank is a strong condition. Indeed, the Cherlin-Zil'ber conjecture states that any infinite simple group \(G\) of finite Morley rank should be an algebraic group over an algebraically closed field. This conjecture, in its full generality, is open [compare \textit{A. Borovik} and \textit{A. Nesin}, Groups of finite Morley rank, Oxford Univ. Press (1994; Zbl 0816.20001)]. The special case where \(G\) is an isotropic algebraic group was established recently by \textit{L. Kramer}, \textit{G. Röhrle} and \textit{K. Tent} [J. Algebra 216, No. 1, 77-85 (1999; Zbl 0935.20031)]. In the paper under review, the authors prove the Cherlin-Zil'ber conjecture for groups \(G\) which contain a spherical BN-pair of Tits rank at least 3, or a spherical Moufang BN-pair of Tits rank 2. The proof is achieved by first classifying all spherical Moufang buildings of finite Morley rank; here most of the work has to be done for the case of Tits rank 2, i.e., for generalized polygons [compare \textit{H. Van Maldeghem}, Generalized polygons, Birkhäuser (1998; Zbl 0914.51005)]. The classification of the groups is then obtained as a consequence of the classification of their underlying geometries. This paper settles the Cherlin-Zil'ber conjecture for a large class of groups. Note that assuming the existence of a BN-pair is natural, because all simple algebraic groups have a BN-pair [see \textit{J. Tits}, Buildings of spherical type and finite BN-pairs, Lect. Notes Math. 386 (1974; Zbl 0295.20047)].
- Ghent University Belgium
- University of Würzburg Germany
generalized polygons, infinite simple groups, spherical BN-pairs, Model-theoretic algebra, algebraic groups, Simple groups, Morley rank, simple algebraic groups, spherical Moufang buildings, Linear algebraic groups over arbitrary fields, Groups with a \(BN\)-pair; buildings, Buildings and the geometry of diagrams, Cherlin-Zil'ber conjecture, Applications of logic to group theory, Classification theory, stability, and related concepts in model theory
generalized polygons, infinite simple groups, spherical BN-pairs, Model-theoretic algebra, algebraic groups, Simple groups, Morley rank, simple algebraic groups, spherical Moufang buildings, Linear algebraic groups over arbitrary fields, Groups with a \(BN\)-pair; buildings, Buildings and the geometry of diagrams, Cherlin-Zil'ber conjecture, Applications of logic to group theory, Classification theory, stability, and related concepts in model theory
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