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

The residually weakly primitive geometries of J2J_2

We announce the end of the classification of all firm and residually connected geometries satisfying the conditions $(IP)_2$ and $(2T)_1$ and on which the Hall-Janko group $J_2$ acts flag-transitively and residually weakly primitively.We state some facts regarding the results.The complete list of geometries is available as a supplement to this paper [9]

On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group McL

We study the homotopy relation between the standard 2-local geometry and the Bouc complex for the sporadic finite simple group McL.Comment: 8 pages, 2 tables, final version to appear in Archiv der Mathemati

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $n\leq 1300$ vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of $n$.Comment: 18 pages, LaTe

On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p^3, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.Comment: 22 page

Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page