Epimorphic images of simplicial Coxeter groups and some associated hyperbolic manifolds

Abstract

In this work torsion-free normal subgroups of the Lannér groups Γ_i are studied.  All torsion-free normal subgroups of the orientation-preserving subgroups Γ_i⁺ whose factor groups have the form L₂(q), where q = pⁿ and p is a prime, are classified.  In the case of each group, some examples of manifolds are constructed and their homology is computed.  Minimal index torsion free subgroups of each Lannér group are also constructed.

Computational techniques are developed to construct complete lists of conjugacy classes of subgroups of low index in these groups.  These lists are then used to test the theoretical results proved in this thesis and also to search for specific subgroups. Computational techniques are also developed to calculate the action of the isometries of these manifolds М on their homology groups.  This give H₁(М) the structure of an Isom(М)-module, which allows for the construction of arbitrarily large manifolds exhibiting a high degree of symmetry.

The computational techniques developed in this work are applied to the 4-dimensional Coxeter group [5,3,3,3] and a detailed study of the low index subgroups of this group has been implemented.  The existence of torsion free subgroups of index 115200 is established and a possible approach towards determining the minimal index torsion-free subgroups of this group is outlined.

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