A quasi Curtis–Tits–Phan theorem for the symplectic group (original) (raw)
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A Curtis–Tits–Phan theorem for the twin-building of type
Journal of Algebra, 2009
The Curtis-Tits-Phan theory as laid out originally by Bennett and Shpectorov describes a way to employ Tits' lemma to obtain presentations of groups related to buildings as the universal completion of an amalgam of low-rank groups. It is formulated in terms of twin-buildings, but all concrete results so far were concerned with spherical buildings only. We describe an explicit flip-flop geometry for the twin-building of type A n−1 associated to k[t, t −1 ] on which a unitary group SU n (k[t, t −1 ], β), related to a certain non-degenerate hermitian form β, acts flag-transitively and obtain a presentation for this group in terms of a rank-2 amalgam consisting of unitary groups. This is the most natural generalization of the original result by Phan for the unitary groups.
Send proofs to: A quasi Phan-Curtis-Tits theorem for the symplectic group
2008
We obtain the symplectic group Sp(V) as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let Sp(V) act flag-transitively on the geometry of maximal rank subspaces of V. We show that this geometry and its rank ≥ 3 residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits ’ lemma. This provides a new way of recognizing the symplectic groups from a small collection of In the revision of the classification of finite simple groups one of the important steps requires one to prove that if a simple group G (the minimal counterexample) contains a certain amalgam of subgroups that one normally finds in a known simple group H then G is isomorphic to H. A geometric approach to recognition theorems was
Realizations and properties of 333-spherical Curtis-Tits Groups and Phan groups
arXiv: Group Theory, 2016
In this note we establish the existence of all Curtis-Tits groups and Phan groups with 333-spherical diagram as classified previously and investigate some of their geometric and group theoretic properties. Whereas it is known that orientable Curtis-Tits groups with spherical or non-spherical and non-affine diagram are almost simple, we show that non-orientable Curtis-Tits groups are acylindrically hyperbolic and therefore have infinitely many infinite-index normal subgroups. However, we also provide concrete examples of non-orientable Curtis-Tits groups whose quotients are finite simple groups of Lie type.
Curtis-Tits groups generalizing Kac-Moody groups of type A_n
2011
In a previous paper we define a Curtis-Tits group as a certain generalization of a Kac-Moody group. We distinguish between orientable and non-orientable Curtis-Tits groups and identify all orientable Curtis-Tits groups as Kac-Moody groups associated to twin-buildings. In the present paper we construct all orientable and non-orientable Curtis-Tits groups with diagram A_n over a field F. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfel'd' s construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The non-orientable ones are related to q-CCR algebras in physics and have symplectic, orthogonal and unitary groups as quotients.
On the local recognition of finite metasymplectic spaces
Journal of Algebra, 1989
This paper is concerned with the local recognition of certain graphs and geometries associated with exceptional groups of Lie type. The local approach to geometries is inspired by group theory. Finite simple groups are often characterized by local information, for example, the fusion pattern of involutions centralizing a given involution. The main results here, although of a geometric nature, are a contribution to obtaining a characterization of a group of exceptional Lie type by the fusion pattern of root subgroups centralizing a given root subgroup.
Curtis-Tits groups generalizing Kac-Moody groups of type widetildeAn\widetilde{A}_nwidetildeAn
2009
In a previous paper we define a Curtis-Tits group as a certain generalization of a Kac-Moody group. We distinguish between orientable and non-orientable Curtis-Tits groups and identify all orientable Curtis-Tits groups as Kac-Moody groups associated to twin-buildings. In the present paper we construct all orientable and non-orientable Curtis-Tits groups with diagram widetildeAn\widetilde{A}_nwidetildeAn over a field mathbbF{\mathbb F}mathbbF. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfel'd' s construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The non-orientable ones are related to q-CCR algebras in physics and have symplectic, orthogonal and unitary groups as quotients.
Simple groups of finite morley rank and Tits buildings
Israel Journal of Mathematics, 1999
THEOREM A: If ~3 is an infinite Moufang polygon of finite Morley rank, then ~3 is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ~3 is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank 2 is either PSL3(K), PSp4(K) or G2(K) for some algebraically closed field K. 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.
The character table of a finite group is a very powerful tool to study the groups and to prove many results. Any finite group is either simple or has a normal subgroup and hence will be of extension type. The classification of finite simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism First of all, I thank ALLAH for his Grace and Mercy showered upon me. I heartily express my profound gratitude to my supervisor, Professor Jamshid Moori, for his invaluable learned guidance, advises, encouragement, understanding and continued support he has provided me throughout the duration of my studies which led to the compilation of • the Postgraduate Diploma project at AIMS-Cape Town 2006, • the MSc in Mathematics at the University of KwaZulu-Natal 2009, • this PhD thesis and hopefully I aim to continue with Prof J. Moori for a Postdoctoral research. I will be always indebted to him for introducing me to this fascinating area of Mathematics and creating my interest in Group Theory. Professor Moori is a unique encyclopaedia and I have learnt so much from him, not only in the academic orientation, but in various walks of life. May ALLAH gives him the power to enrich furthermore this interesting domain of Mathematics, and in general to advance the wheel of life forward. I lovingly thank my precious wife Muna, who supported me each step of the way and without her help and encouragement it simply never would have been possible to finish this work.