Reference for tetrahedral Coxeter group

Question

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.

Answer 1

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 in Matrix Realizations of the Hyperbolic Tetrahedral Groups in ${PSL}_2(\mathbb C)$.

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

Answer 2

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 subdivision, 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.

Answer 3

I recently ran across the paper "3D Farey graph, lambda lengths and $SL_2$-tilings" by Anna Felikson, Oleg Karpenkov, Khrystyna Serhiyenko and Pavel Tumarkin. Let $\sigma$ be a primitive $6$-th root of unity.

The authors of this paper identify the Coxeter group in question with $\Gamma \rtimes \langle \tau \rangle$, where $\Gamma \subset \text{PSL}_2(\mathbb{Z}[\sigma])$ is the subgroup of matrices which are the identity modulo $1+\sigma$ and $\tau$ acts on $\text{PSL}_2(\mathbb{Z}[\sigma])$ by complex conjugation. The Coxeter group acts on hyperbolic $3$-space, providing a tiling of $\mathbb{H}^3$ by ideal tetrahedra with vertices at infinity. The authors show that these vertices are in bijection with $\mathbb{P}^1(\mathbb{Q}[\sigma])$ and they give conditions for $2$, $3$ or $4$ points of $\mathbb{P}^1(\mathbb{Q}[\sigma])$ to form the vertices of an edge/triangle/tetrahdron in this tiling. They then show how to compute Penner's $\lambda$-lengths for geodesics between these vertices.

All of this is very analogous to the way that the free Coxeter group on $3$ generators is usually identified with a congruence subgroup of $\text{PSL}_2(\mathbb{Z}) \rtimes \langle \tau \rangle$, and its cusps are identified with $\mathbb{P}^1(\mathbb{Q})$.

Another useful term that I learned from this paper is that this group, or maybe the group $PSL_2(\mathbb{Z}[\sigma])$ (I'm not quite clear), is called the Bianchi group $\text{Bi}(3)$.