Arithmetic Fuchsian group

A Fuchsian group $\Gamma$ is arithmetic if it is commensurable up to conjugacy in $\operatorname{PSL}_2(\mathbb R)$ with the projective image of $\mathcal O^1$ , for the following data:

a totally real number field $F$ ;
a quaternion algebra $B/F$ , split at exactly one real embedding and ramified at the others;
an order $\mathcal O\subset B$ over the ring of integers of $F$ , with norm-one group $\mathcal O^1$ .

The image uses a splitting $B\otimes_{F,\sigma}\mathbb R\cong M_2(\mathbb R)$ at the distinguished real embedding $\sigma$ , followed by quotienting by $\{\pm I\}$ .

Examples and scope

Taking $F=\mathbb Q$ , $B=M_2(\mathbb Q)$ , and $\mathcal O=M_2(\mathbb Z)$ gives $\operatorname{PSL}_2(\mathbb Z)$ .

Such groups have finite hyperbolic covolume. An arbitrary infinite-index subgroup of one need not be arithmetic in this sense. A definite rational quaternion algebra has no split real place and does not yield a Fuchsian lattice by this construction.

References

John Voight, Quaternion Algebras, §38.3, Definition 38.3.4, Proposition 38.3.8, and §38.3.10; see also §38.4 for compactness.