Sergey Shpectorov

On scale embeddings of graphs into hypercubes

Abstract We investigate graphs that are isometrically embeddable into the metric space l1.

A Phan-type theorem for Sp(2n,q)

In 1977 Kok-Wee Phan published a theorem (see [4]) on generation of the special unitary group SU(n+ 1, q2) by a system of its subgroups isomorphic to SU(3, q2). This theorem is similar in spirit to the famous Curtis–Tits theorem. In fact, both the Curtis–Tits theorem and Phan’s theorem were used as principal identification tools in the classification of finite simple groups. The proof of Phan’s theorem given in his 1977 paper is somewhat incomplete. This motivated Bennett and Shpectorov [1] to revise Phan’s paper and provide a new and complete proof of his theorem. They used an approach based on the concepts of diagram geometries and amalgams of groups. It turned out that Phan’s configuration arises as the amalgam of rank two parabolics in the flag-transitive action of SU(n + 1, q2) on the geometry of nondegenerate subspaces of the underlying unitary space. This point of view leads to a twofold interpretation of Phan’s theorem: its complete proof must include (1) a classification of related amalgams; and (2) a verification that—apart from some small exceptional cases—the above geometry is simply connected. These two parts are tied together by a lemma due to Tits, that implies that if a group G acts flag-transitively on a simply connected geometry then the corresponding amalgam of maximal parabolics provides a presentation for G, see Proposition 7.1. The Curtis–Tits theorem can also be restated in similar geometric terms. Let G be a Chevalley group. Then G acts on a spherical building B and also on the corresponding twin building B = (B+,B−, d∗). (Here B+ ∼= B ∼= B− and d∗ is a codistance between

Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

Let ? be a finite thick dual polar space of rank 3. We say that a hyperplane H of ? is locally singular (respectively, quadrangular or ovoidal) if H?Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of ?. 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 ? or ? is the dual of Q(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 Q(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 ?\H induced by ? on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with ?\H flag-transitive are classified.

On the definition of saturated fusion systems

Abstract We propose a simplification of the definition of saturation for fusion systems over p-groups and prove the equivalence of our definition with that of Broto, Levi, and Oliver.

A GAP Package for Braid Orbit Computation and Applications

Let G be a finite group. By Riemanns Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus, many problems on algebraic curves require the computation of braid orbits. In this paper, we describe an implementation of this computation. We discuss several applications, including the classification of irreducible familiesof indecomposable rational functions with exceptional monodromy group.

The flag-transitive tilde and Petersen-type geometries are all known

We announce the classification of two related classes of flag-transitive geometries. There is an infinite family of such geometries, related to the nonsplit extensions

Primitive axial algebras of Jordan type

3^{[{n\atop 2}]_{_2}}\cdot \SP_{2n}(2)

Recognition of the l 1 -graphs with complexity O ( nm ), or football in a hypercube

, and twelve sporadic examples coming from the simple groups

Geometries for Sporadic Groups Related to the Petersen Graph: II

M_{22}

Distance-regular graphs the distance matrix of which has only one positive eigenvalue

,