Universal Covers of Geometries of Far Away Type

Journal of Algebraic Combinatorics, 18, 211–243, 2003 c 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
Universal Covers of Geometries of Far Away Type ANTONIO PASINI University of Siena, Department of Mathematics, Via del Capitano 15, 53100 Siena, Italy Received March 7, 2002; Revised January 24, 2003; Accepted January 26, 2003 Abstract. The geometries studied in this paper are obtained from buildings of spherical type by removing all chambers at non-maximal distance from a given element or flag. I consider a number of special cases of the above construction chosen among those which most frequently appear in the literature, proving that the resulting geometry is always simply connected but for three cases of small rank defined over GF(2) and GF(4). I also compute the universal cover in those exceptional cases. Keywords: buildings, universal covers, embeddings, binary codes 1. Introduction Geometries obtained from buildings of spherical type by removing all elements at nonmaximal distance from a given element or a flag, are met in the context of many interesting characterizations and classifications. Many of them also appear in connection with embeddings of buildings of spherical type, as affine expansions of some of those embeddings (see [16]). As shown in [16], the hull of an embedding corresponds to the universal cover of the expansion of that embedding. In particular, an embedding is its own hull if and only if its expansion is simply connected. In this paper I consider a number of special cases of the construction sketched above, proving that nearly all of the geometries obtained in those cases are simply connected. It is likely that the same conclusion holds for more families of ‘far away’ geometries, different from those studied here. In fact, I have only considered those families that either are related to some of the embeddings discussed in [16] or include examples that have been investigated by some authors in some contexts. Actually, my selection misses one family which however, according to the above criteria, deserved to be studied, namely the case of the subgeometry of a building of type F4 far from a given point or symp. I have not considered it simply because I couldn’t find the right way to treat it. We follow [13] for basic notions and general results on geometries and Tits [18] for buildings. In particular, according to [13], we assume all geometries to be residually connected and firm, by definition. We refer to chapters 8, 11 and 12 of [13] for m-covers, m-quotients and m-simple connectedness, but we are only interested in 2- and (n − 1)-covers in this paper. We recall that the (n − 1)-covers of a geometry of rank n are called topological covers in [13], but many authors simply call them covers. We too do so in this paper. Accordingly, in the sequel, the 212 PASINI universal cover of a geometry
of rank n is its universal (n − 1)-cover and we say that
is simply connected if it is (n − 1)-simply connected. 1.1. The geometry far from a flag Suppose  is a thick building of connected spherical type and rank at least 2. It is well known that, given a flag F
= ∅ and a chamber C of , there is a unique chamber C F ⊇ F at minimal distance from C (Tits [18]). We denote the distance between C and C F by d(C, F). For every nonempty flag X , the distance d(X, F) from X to F is the minimal distance d(C, F) from F to a chamber C ⊇ X . We say that a flag X is far from F if d(X, F) is maximal, compatibly with the types of F and X . We denote by Far (F) the substructure of  formed by the elements far from F, with the incidence relation inherited from  but rectified as follows: two elements x, y of Far (F) are incident in Far (F) if and only if they are incident in  and the flag {x, y} is far from F. As  is thick, the structure Far (F) is firm. It is known that Far (F) is residually connected (whence, it is a geometry) except for a few cases defined over GF(2) (Blok and Brouwer [4]), but none of those exceptional cases will be met in this paper. In the sequel we call Far (F) a geometry of far away type, also a far away geometry, for short. Before to state the results of this paper, we mention a few examples, focusing on simple connectedness. Example 1.1 Suppose  is a non-degenerate projective geometry of dimension n ≥ 3 and let A be a hyperplane or a point of . Then Far (A) is an affine geometry or the dual of an affine geometry. In any case, Far (A) is simply connected. Example 1.2 With  as in Example 1, let F be a point-hyperplane flag of . Then Far (F) is an affine-dual-affine geometry as in Van Nypelseer [19]. It follows from the main result of [19] that Far (F) is simply connected. Example 1.3 Let  be a thick polar space of rank n > 2 and p a point of . Then Far ( p) is an affine polar space (Cohen and Shult [7]). Affine polar spaces are simply connected (Pasini [12]; also Cuypers and Pasini [8] and [13, Proposition 12.50]). So, Far ( p) is simply connected. A similar conclusion holds when  is a building of type Dn and p is an element of  corresponding to a point of the polar space  associated to . Indeed, the subgeometry of  corresponding to Far ( p) is an affine polar space and, as recalled above, affine polar spaces are simply connected. 1.2. Main results The geometries Far (A), Far (F) and Far ( p) of Examples 1.1, 1.2 and 1.3 are simply connected. In this paper we obtain the same conclusion in a number of other cases. Explicitly, the following are the results we shall prove. UNIVERSAL COVERS OF GEOMETRIES 213 Theorem 1.1 Let  be a non-degenerate projective geometry of dimension n ≥ 3 and A an element of  other than a point or a hyperplane. Then Far (A) is simply connected. Theorem 1.2 Let  be a thick polar space of rank n ≥ 3 and A a maximal singular subspace of . Namely t(A) = n − 1, where the nonnegative integers 0, 1, 2, . . . , n − 1 are taken as types, as follows: Then Far (A) is simply connected, except when n = 3 and  is either the symplectic variety S(5, 2) of P G(5, 2) or the hermitean variety H(5, 4) of P G(5, 4). Theorem 1.3 For n ≥ 4, let  be a thick building of type Dn and A an element corresponding to a maximal singular subspace of the polar space associated to . That is, t(A) = + or −, where the nonnegative integers 0, 1, 2, . . . , n − 3 and the symbols + and − are taken as types, as follows: Then Far (A) is simply connected. Theorem 1.4 For n ≥ 4, let  be a thick building of type Dn and F a flag of  of type {+, −}, with types as in Theorem 1.4. Then Far (F) is simply connected except when n = 4 and  is defined over GF(2). Theorem 1.5 Let  be a thick building of type E 6 , with types 0, 1, 2, 3, 4, 5 as follows: Then Far ( p) is simply connected for every 0-element p. A few small cases are not covered by the previous theorems: Theorem 1.2 misses the cases of  = S(5, 2) and  = H(5, 4). Theorem 1.5 misses the case of  = D4 (2), the building of (...truncated)