Arithmetic hyperbolic 3-manifold

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$ by an arithmetic Kleinian group. These manifolds include some particularly beautiful or remarkable examples.

A quaternion algebra over a field ${\displaystyle F}$ is a four-dimensional central simple ${\displaystyle F}$-algebra. A quaternion algebra has a basis ${\displaystyle 1,i,j,ij}$ where ${\displaystyle i^{2},j^{2}\in F^{\times }}$ and ${\displaystyle ij=-ji}$.

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