regular polytopes Thin regular geometries Almost simple groups of PSL(2, q) type Our paper deals with the classification of abstract regular polytopes for almost simple groups with socle PSL(2, q). We consider all almost simple groups PSL(2, q) ≤ G ≤ PΓL(2, q) and determine the maximal rank of string C-group representations for G, i.e. the maximal rank of an abstract regular polytope with automorphism group G, as well as the existence of string C-group representations of lower ranks. Similar results have already been obtained by various authors in the cases G ∼ = PSL(2, q) and G ∼ = PGL(2, q) and they are summarized in this paper.

We determine all firm and residually connected rank 2 geometries on which PSL(2, q) acts flag-transitively, residually weakly primitively and locally two-transitively, where one of the maximal parabolic subgroups is isomorphic to E q : (q−1) (2,q−1) , where E q denotes an elementary abelian group of order q, or D 2n(q) , the dihedral group of order 2n(q) where n(q) := (q±1) gcd(2,q−1) for some prime-power q.

We determine all firm and residually connected rank 2 geometries on which PSL(2, q) acts flag-transitively, residually weakly primitively and locally two-transitively, in which one of the maximal parabolic subgroups is isomorphic to A 4 , S 4 , A 5 , PSL(2, q) or PGL(2, q), where q divides q, for some prime-power q.

ABSTRACT. It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge-transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.

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