I would like to understand compact hyperbolic manifolds a bit better. Hyperbolic manifolds may be described as quotients of hyperbolic space by a torsion-free, discrete subgroup (lattice) of the isometry group $\mathrm{PO}(n,1)$.
Let us start in dimension 2. I would like to study a compact quotient of the hyperbolic plane $\mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2)$ as a toy example. That means I need a cocompact lattice $\Gamma\subset\mathrm{SL}(2,\mathbb{R})$.
Does anyone know an example?
By §XII.1, Thm. 3 in Lang's book $SL_2(\mathbb{R})$, $\Gamma$ may not contain nontrivial unipotent elements. That means it cannot simply be an extension of $\mathrm{SL}(2,\mathbb{Z})$ because this one already contains, for example, the matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$.