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 possibly its index two subgroup, sources seem to differ), is called the Bianchi group $\text{Bi}(3)$.

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 $\text{Bi}(3)$.

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 $\text{PSL}_2(\mathbb{Z}[\sigma]) \rtimes \langle \tau \rangle$, where $\tau$ 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 possibly its index two subgroup, sources seem to differ), is called the Bianchi group $\text{Bi}(3)$.

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 possibly its index two subgroup, sources seem to differ), is called the Bianchi group $\text{Bi}(3)$.