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

Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

A new near octagon and the Suzuki tower

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.Comment: 24 pages, revised version with added remarks and reference