Epimorphic images of the $$[5,3,5]$$ Coxeter group

Hyperbolic manifolds and tessellations of type {3,5,3} associated with L_2(q)

“…On the basis of this evidence we conjecture that Ω + /K ∼ = L 2 (q) × C 2 whenever F q > F . A similar phenomenon for the tetrahedral group [5,3,5] + is observed in [19], where the analogues of Theorems 1.1 and 1.2 are proved for this group.…”

“…II §9, §10], for instance). In particular, when Γ = [3,5,3] the corresponding kernels are the normal subgroups K appearing in Theorem 1.1, or intersections of them for a given q; a similar remark applies to the discussion of Γ = [5,3,5] in [19]. The advantage of our more grouptheoretic approach is that it allows one to show that all the normal subgroups with a given quotient of type L 2 have been obtained.…”

“…Leemans and Schulte [21] have shown that the only other regular abstract polytope of rank greater than 3 with symmetry group L 2 (q) is the 57-cell (or pentacontaheptachoron), with 57 hemidodecahedral facets. This has symmetry group L 2 (19), arising as a quotient of the Coxeter group [5,3,5]; quotients isomorphic to L 2 (q) of this and related groups have been studied more generally by Gradolato and Zimmermann [16], by Paoluzzi [24], and by two of the present authors [19].…”

“…By inspection of the cases q ≤ 59, using Dickson's classification [6, Ch.XII] of the subgroups of L 2 (q), the only examples of this phenomenon (with a non-identity complement) are those described above with q = 11, 29 or 59, so these are the only cases where M is a regular covering of a manifold consisting of a single icosahedron. There are, however, other examples where a quotient manifold M/S consists of a small number of icosahedra: for instance, taking S to be a Sylow 19-subgroup in L 2 (19) or a Sylow 17-subgroup in L 2 (2 4 ) yields tessellations with three or four cells respectively.…”

“…In this paper we determine all the normal subgroups (necessarily torsion-free) K of ∆ with ∆/K ∼ = L 2 (q) for any q, together with their normalisers N(K), and we describe the resulting isometry groups N(K)/K and symmetry groups N Γ (K)/K. Some of the methods used here are similar to those applied in [19] to the Coxeter group [5,3,5] and its normaliser, associated with the dodecahedral tessellation {5, 3, 5} of H 3 .…”