Extended F_4-buildings and the Baby Monster

The Baby Monster group B acts naturally on a geometry E(B) with diagram c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2, and (b) three elements of type 1 are pairwise incident to common elements of type 2 iff they are incident to a common element of type 5. It is shown that E(B) is the only (non-necessary flag-transitive) c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first characterization of B in terms of an incidence geometry, similar in vein to one known for classical groups acting on buildings. Further, it is shown that E(B) contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2 with flag-transitive automorphism groups are constructed. A complete list of possibilities for the isomorphism type of the subgraph induced by the common neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t) satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Natural representations of some tilde and petersen type geometries

We determine the universal natural representations (embeddings) of certain sporadic geometries with three points per line.</p

On scale embeddings of graphs into hypercubes

We investigate graphs that are isometrically embeddable into the metric space l1.</p

Classification of certain types of tilde geometries

We show that a certain class of diagram geometries called tilde geometries of symplectic type is simply connected. Here we prove that the corresponding amalgam is uniquely determined. The result then follows from Ivanov and Shpectorov (Geom. Dedicata45 (1993), 1-23).</p

Uniform hyperplanes of finite dual polar spaces of rank 3

Let Delta be a finite thick dual polar space of rank 3. We say that a hyperplane H of Delta is locally singular (respectively, quadrangular or ovoidal) if H boolean AND Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Delta. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Delta or Delta is the dual of L(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in L(6, 2) and H (5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Delta H induced by Delta on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Delta H flag-transitive are classified. (C) 2001 Academic Press

P-Geometries of Rank 3

Under minor extra assumptions, we classify P-geometries of rank 3. Previously, a classification was known only for flag-transitive P-geometries.</p

The P-geometry for M₂₃ has no Non-trivial 2-coverings

It is shown that the P-geometry related to the group M23 is 2-simply connected.</p

Geometry of sporadic groups

The second in a two-volume set, for researchers into finite groups, geometry and algebraic combinatorics

Amalgams determined by locally projective actions

A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph γ of valency 2n-1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)γ(x) is the linear group SLn(2) ≅ Ln(2) in its natural doubly transitive action and (iii) {t, G{x, y}] ≤ O2(G(x) γ G{x, y}) for some t ε G{x, y} \ G(x). D. Ž. Djoković and G. L. Miller [DM80], used the classical Tutte's theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov's theorem [Tr01] to extend the classification to the case n ≥ 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J 4 and BM. For each of the exceptional amalgam n = 3, 4 or 5.</p

The association schemes of dual polar spaces of type ²A_2d-1(p^f) are characterized by their parameters if d≥3

It is shown that the cliques in a distance-regular graph Γ whose parameters are that of the dual polar space graph of type 2A2d-1(pf) have size pf + 1. This means that Γ is the point graph of a regular near 2d-gon. If d = 2, we obtain the well-known result due to P.J. Cameron, J.M. Goethals, and J.J. Seidel that a pseudogeometric graph with parameters of the point graph of a generalized quadrangle of type (q,q2) is geometric. If d≥3, then some results due to P.J. Cameron, E.E. Shult, A. Yanushka, and J. Tits on near 2d-gons imply that Γ coincides with the dual polar space graph of type 2A2d-1(pf).</p