# Reference for tetrahedral Coxeter group

Let G be the group with 4 generators, each of order 2, such that the product of any 2, say ab, has order 3 (i.e., ababab=e). That is, this is an infinite reflection group with Coxeter diagram a tetrahedron. I am looking for references for this group...

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In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.

Here are some:

Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds" has the hyperbolic tetrahedral groups arranged into commensurability classes in the Appendix (Chapter 13) and a longer discussion of about the groups in section 4.7.

Representations of the orientation preserving subgroups of these Coxeter groups into $PSL(2,\mathbb{C})$ as computed by Grant Lakeland are available here.

The non-hyperbolic examples of these Coxeter groups are classified according to the geometries of their quotients in William Dunbar's paper "Geometric Orbifolds."

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The group itself shall be the group generated by reflections in the sides of a regular ideal tetrahedron, whose dihedral angles are all $\pi/3$. For a reference, there are many Coxeter diagrams listed in a paper by Johnson, Kellerhals, Ratcliffe and Tschantz, called "The size of a hyperbolic simplex". Please mind the fact that you may have to look at the barycentric subdivision of your simplex before you find its counterpart (a simplex from the ubdivision, which is an orthoscheme) in their table. Definitely, the ideal simplex reflection group contains interesting subgroups, which are manifold groups. There should be the eight-knot group in there, since the figure-eight complement comes from glueing two regular ideal tetrahedra. I suppose that Neil's references shed more light on this kind of questions. Hope my reply describes in more geometric detail the group you were interested in.