On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group Co3

We study the homotopy relation between the standard 2-local geometry and the Bouc complex for the sporadic group Co3. We also give a result concerning the relative projectivity of the reduced Lefschetz module associated to the aformentioned 2-local geometry.Comment: 20 page

Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set \cFl (\cF, E) of generalized flags E$--commensurable with a fixed generalized flag$\cF$in$V$has a natural structure of an ind--variety. In the case when$V$is the standard representation of$G = SL(\infty)$, all homogeneous ind--spaces$G/P$for parabolic subgroups$P$containing a fixed splitting Cartan subgroup of$G$, are of the form$\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for$SO(\infty)$and$Sp(\infty)$. As an application of the construction, we compute the Picard group of$\cFl (\cF, E)$(and of its isotropic analogs) and show that$\cFl (\cF, E)$is a projective ind--variety if and only if$\cF$is a usual, possibly infinite, flag of subspaces in$V$

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon \cS, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group.Comment: forgot to upload the appendices in version 1, and this is rectified in version 2. erased cross-ref keys in version 3. Minor revision in version 4 to implement the suggestion by the referee (new section at the end, extended acknowledgment, simpler proof for Lemma 4.2

Groupoids, imaginaries and internal covers

Let $T$ be a first-order theory. A correspondence is established between internal covers of models of $T$ and definable groupoids within $T$. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.Comment: Local improvements; thanks to referee of Turkish Mathematical Journal. First appeared in the proceedings of the Paris VII seminar: structures alg\'ebriques ordonn\'ee (2004/5