322 research outputs found
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
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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