Conway groupoids, regular two-graphs and supersimple designs

A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families of groupoids arise in this way, some associated with 3-transposition groups, which have two additional properties. Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. In this paper, we show each of these properties corresponds to a group-theoretic property on the groupoid and we classify the Conway groupoids and the supersimple designs for which both of these two additional properties hold.Comment: 17 page

The geometry of hyperbolic lines in polar spaces

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have applications in group theory, in the theory of Lie algebras and in graph theory

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