Ralf Gramlich

A Phan-type theorem for Sp(2n,q)

In 1977 Kok-Wee Phan published a theorem (see [4]) on generation of the special unitary group SU(n+ 1, q2) by a system of its subgroups isomorphic to SU(3, q2). This theorem is similar in spirit to the famous Curtis–Tits theorem. In fact, both the Curtis–Tits theorem and Phan’s theorem were used as principal identification tools in the classification of finite simple groups. The proof of Phan’s theorem given in his 1977 paper is somewhat incomplete. This motivated Bennett and Shpectorov [1] to revise Phan’s paper and provide a new and complete proof of his theorem. They used an approach based on the concepts of diagram geometries and amalgams of groups. It turned out that Phan’s configuration arises as the amalgam of rank two parabolics in the flag-transitive action of SU(n + 1, q2) on the geometry of nondegenerate subspaces of the underlying unitary space. This point of view leads to a twofold interpretation of Phan’s theorem: its complete proof must include (1) a classification of related amalgams; and (2) a verification that—apart from some small exceptional cases—the above geometry is simply connected. These two parts are tied together by a lemma due to Tits, that implies that if a group G acts flag-transitively on a simply connected geometry then the corresponding amalgam of maximal parabolics provides a presentation for G, see Proposition 7.1. The Curtis–Tits theorem can also be restated in similar geometric terms. Let G be a Chevalley group. Then G acts on a spherical building B and also on the corresponding twin building B = (B+,B−, d∗). (Here B+ ∼= B ∼= B− and d∗ is a codistance between

Abstract involutions of algebraic groups and of Kac–Moody groups

Abstract Based on the second authors thesis [Horn, Involutions of Kac–Moody groups, TU Darmstadt, 2008], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac–Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [Helminck, Wang, Adv. Math. 99: 26–96, 1993] and [Springer, Algebraic groups and related topics: 525–543, North-Holland, 1984] for algebraic groups and in [Kac, Wang, Adv. Math. 92: 129–195, 1992] for certain Kac–Moody groups, we analyze the filtration studied in [Devillers, Mühlherr, Forum Math. 19: 955–970, 2007] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [Bennett, Gramlich, Hoffman, Shpectorov, Curtis–Phan–Tits theory: 13–29, World Scientific, 2003].

The complete Phan-type theorem for Sp(2n,q)

Abstract The articles [R. Gramlich, C. Homan and S. Shpectorov. A Phan-type theorem for Sp(2n; q). J. Algebra 264 (2003), 358–384.] and [R. Gramlich. Weak Phan systems of type C n . J. Algebra 280 (2004), 1–19.] give a characterization of central quotients of the group Sp(2n, q) for n ≥ 3 and all prime powers q, up to some small cases that are left open. The present article fills in this gap, thus providing the definitive version of the Phan-type theorem for Sp(2n, q).

On the hyperbolic symplectic geometry

The present article provides a new characterization of the geometry on the points and hyperbolic lines of a non-degenerate symplectic polar space. This characterization is accomplished by studying the family of subspaces obtained when considering the polars of all hyperbolic lines.

Local Recognition Of Non-Incident Point-Hyperplane Graphs

Let ℙ be a projective space. By H(ℙ) we denote the graph whose vertices are the non-incident point-hyperplane pairs of ℙ, two vertices (p,H) and (q,I) being adjacent if and only if p ∈ I and q ∈ H. In this paper we give a characterization of the graph H(ℙ) (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSLn(

Opposition in triality

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Local recognition of the line graph of an anisotropic vector space

)≤G≤PGLn(