Abstract
We classify the normal subgroups of the Coxeter group \(\varGamma =[5,3,5]\), and of its even subgroup \(\varGamma ^+\), with quotient isomorphic to a finite simple group \(L_2(q)\). There are infinitely many such normal subgroups of \(\varGamma ^+\), each uniformising a compact orientable hyperbolic \(3\)-manifold tessellated by dodecahedra; we determine the isometry groups of these manifolds and the symmetry groups of their tessellations. By contrast there is a single such normal subgroup of \(\varGamma \), uniformising a compact non-orientable \(3\)-orbifold with isometry group \(PGL_2(19)\).
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Jones, G.A., Long, C.D. Epimorphic images of the \([5,3,5]\) Coxeter group. Math. Z. 275, 167–183 (2013). https://doi.org/10.1007/s00209-012-1129-2
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DOI: https://doi.org/10.1007/s00209-012-1129-2