Corneliu Hoffman

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

Bass-Serre theory and counting rank two amalgams.

An amalgam of groups can be viewed as a Sudoku game inside a group. You are given a set of subgroups and their intersections and you need to decide what the largest group containing such a structure can be. In a recent paper (0907.1388v1) we used Bass-Serre theory of graphs of groups to classify all possible amalgams of Curtis-Tits shape with a given diagram. This note describes the method for general rank two amalgams.

1-Cohomology of simplicial amalgams of groups

We develop a cohomological method to classify amalgams of groups. We generalize this to simplicial amalgams in any concrete category. We compute the non-commutative 1-cohomology for several examples of amalgams defined over small simplices.

A Classification of Curtis-Tits Amalgams

A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that these groups are determined by the local structure arising from their fundamental subgroups of ranks \(1\) and \(2\). This result was later extended to Kac-Moody groups by P. Abramenko and B. Muhlherr and Caprace. Their theorem states that a Kac-Moody group \(G\) is the universal completion of an amalgam of rank two (Levi) subgroups, as they are arranged inside \(G\) itself. Taking this result as a starting point, we define a Curtis-Tits structure over a given diagram to be an amalgam of groups such that the sub-amalgam corresponding to a two-vertex sub-diagram is the Curtis-Tits amalgam of some rank-\(2\) group of Lie type. There is no a priori reference to an ambient group, nor to the existence of an associated (twin-) building. Indeed, there is no a priori guarantee that the amalgam will not collapse. We then classify these amalgams up to isomorphism. In the present paper we consider triangle-free simply-laced diagrams. Instead of using Goldschmidt’s lemma, we introduce a new approach by applying Bass and Serre’s theory of graphs of groups, not to the amalgams themselves but to a graph of groups consisting of certain automorphism groups. The classification reveals a natural division into two main types: “orientable” and “non-orientable” Curtis-Tits structures. Our classification of orientable Curtis-Tits structures naturally fits with the classification of all locally split Kac-Moody groups over fields with at least four elements using Moufang foundations. In particular, our classification yields a simple criterion for recognizing when Curtis-Tits structures give rise to Kac-Moody groups. The class of non-orientable Curtis-Tits structures is in some sense much larger. Many of these amalgams turn out to have non-trivial interesting completions inviting further study.

Curtis–Tits groups of simply-laced type

The classification of Curtis-Tits amalgams with {connected}, triangle free, simply-laced diagram over a field of size at least

On a quasi-Phan theorem for orthogonal groups.

4

Even-dimensional orthogonal groups as amalgams of unitary groups

was completed in~\cite{BloHof2014b}. Orientable amalgams are those arising from applying the Curtis-Tits theorem to groups of Kac-Moody type, and indeed, their universal completions are central extensions of those groups of Kac-Moody type. The paper~\cite{BloHof2014a} exhibits concrete (matrix) groups as completions for all Curtis-Tits amalgams with diagram

Cross Characteristic Projective Representations for Some Classical Groups

\widetilde{A}_{n-1}

A Quasi Curtis-Tits-Phan theorem for the symplectic group

. For non-orientable amalgams these groups are symmetry groups of certain unitary forms over a ring of skew Laurent polynomials. In the present paper we generalize this to all amalgams arising from the classification above and, under some additional conditions, exhibit their universal completions as central extensions of twisted groups of Kac-Moody type.

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: general simple connectedness

We construct a Phan-like geometry for the orthogonal group SO +(V) and prove that, under some conditions on V this geometry is connected and simply connected. This provides a very small amalgam presentation for the orthogonal group.