Dimitri Leemans

Groups of type L 2 (q) acting on polytopes

We prove that if G is a string C-group of rank 4 and G ∼ L2(q) with q a prime power, then q must be 11 or 19. The polytopes arising are

Polytopes of high rank for the alternating groups

There is a well-known correspondence between abstract regular polytopes and string C-groups. In this paper, for each d>=3, a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n>=9), or equivalently an abstract regular d-polytope, is produced with automorphism group Alt(n). A method that extends the CPR graph of a polytope to a different CPR graph of a larger (or possibly isomorphic) polytope is used to prove that various groups are themselves string C-groups.

Two atlases of abstract chiral polytopes for small groups

We construct chiral abstract polytopes in two different ways. Firstly we seek them as quotients of regular polytopes arising from the Atlas of Small Regular Polytopes ( http://www.abstract-polytopes.com/atlas/ ); the resulting atlas of chiral polytopes atlas is available on the website http://www.abstract-polytopes.com/chiral/ . Secondly, for each almost simple group Γ such that S ≤ Γ ≤ Aut( S ) where S is a simple group and Γ is a group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract chiral polytopes on which Γ acts regularly. The results have been obtained using a series of Magma programs. All these polytopes are made available on the third authors website, at http://math.auckland.ac.nz/~dleemans/CHIRAL/ .

Almost simple groups of Suzuki type acting on polytopes

Let S = Sz(q), with q ≠ 2 an odd power of two. For each almost simple group G such that S < G < Aut(S), we prove that G is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For G = Sz(q), we show that there is always at least one abstract regular polytope P such that G = Aut(P). Moreover, if P is an abstract regular polytope such that G = Aut(P), then P is a polyhedron.

The Residually Weakly Primitive Geometries of the Janko Group J1

We classify all firm, residually connected coset geometries, on which the group J 1 acts as a flag-transitive automorphism group fulfilling the primitiv-ity condition RWPRI: For each flag F, its stabilizer acts primitively on the elements of some type in the residue ΓF- We demand also that every residue of rank two satisfies the intersection property.

All Alternating Groups

Using the correspondence between abstract regular polytopes and string C-groups, in a recent paper [M. E. Fernandes, D. Leemans, and M. Mixer, J. Combin. Theory Ser. A, 119 (2012), pp. 42–56], we constructed an abstract regular polytope of rank r, for each

A_n

r \geq 3

with

, with automorphism group isomorphic to

n\geq12

A_{2r+3}

Have Polytopes of Rank

when r is odd, and