Let $k$ be a number field with one complex place and let $B$ be a quaternion algebra defined over $k$ which ramifies at every real place of $k$ (and perhaps a finite place as well - the number of ramified primes must be even). Let $\mathcal{O}$ be an order of $B$ and $\mathcal{O}^1$ its elements of reduced norm $1$. Choosing an embedding $k\hookrightarrow \mathbb{C}$ induces an embedding $B\hookrightarrow M_2(\mathbb{C})$, which in turn restricts to an embedding $\Psi: \mathcal{O}^1\hookrightarrow SL_2(\mathbb{C})$. A group which is commensurable with $\Psi(\mathcal{O}^1)$ is called arithmetic. An arithmetic group is cocompact if and only if the quaternion algebra $B$ is a division algebra.

Let $k$ be a number field with one complex place and let $B$ be a quaternion algebra defined over $k$ which ramifies at every real place of $k$ (and perhaps some finite places as well - Hilbert reciprocity implies that the total number of ramified places must be even). Let $\mathcal{O}$ be an order of $B$ and $\mathcal{O}^1$ its elements of reduced norm $1$. Choosing an embedding $k\hookrightarrow \mathbb{C}$ induces an embedding $B\hookrightarrow M_2(\mathbb{C})$, which in turn restricts to an embedding $\Psi: \mathcal{O}^1\hookrightarrow SL_2(\mathbb{C})$. A group which is commensurable with $\Psi(\mathcal{O}^1)$ is called arithmetic. An arithmetic group is cocompact if and only if the quaternion algebra $B$ is a division algebra (by Wedderburn's Theorem this is equivalent to saying that the set of places of $k$ which ramify in $B$ is nonempty).