Rieuwert J. Blok

Spanning point-line geometries in buildings of spherical type

We consider the point-line geometries that arise as a shadow space in a spherical building with a diagram of type An, Bn, Cn, Dn or En, and determine in which cases the geometry is spanned by the set of points on an apartment. It turns out that this happens precisely in the cases corresponding to a minimal weight.

The Geometry Far from a Residue

We show that in general the subgeometry of a building of spherical type induced by all objects in general position with respect to a given residue is a residually connected geometry with a Buekenhout-Tits diagram resembling the original diagram, but with certain strokes replaced by the corresponding ‘affine’ strokes. The exceptions (where connectedness fails) are discussed in some detail.

On absolutely universal embeddings

It is well known that, given a point-line geometry Γ and a projective embedding e: Γ → PG(V), if dim(V) equals the size of a generating set of Γ, then e is not derived from any other embedding. Thus, if Γ admits an absolutely universal embedding, then e is absolutely universal. In this paper, without assuming the existence of any absolutely universal embedding, we give sufficient conditions for an embedding e as above to be absolutely universal.

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.

Partial orders generalizing the weak order on Coxeter groups

We define a new family of partial orders generalizing the weak order on Coxeter groups called T-orders, where T is a set of reflections determining the covers in this order. We show that the Grassmann and Lagrange orders on the Coxeter groups of type An and Bn introduced by Bergeron and Sottile are in fact T-orders. These partial orders were used to compute certain products in the cohomology ring of the flag manifolds associated to the complex Chevalley groups of these types. We exhibit T-orders generalizing these orders to partial orders for the Coxeter groups of type Dn, E6, and E7.

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.

A thin near hexagon with 50 points

We show the existence and uniqueness of a thin near hexagon which has 50 points and an affine plane of order 3 as a local space.

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

Coxeter–Chein Loops

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