I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples.
I wrote down that a group $\Gamma$ generated by reflections in a Coxeter polyhedron $P \subset H^n$ was examples of such a lattice for any $n$. Moreover, I said that if $P$ had an ideal vertex, the group acted noncocompactly, and if it had no ideal vertices, the group acted cocompactly (since $H^n / \Gamma$ is isometric to $P$).
In perusing our old class notes, however, I came across the following two results:
Theorem (Vinberg) There do not exist compact Coxeter polyhedra in $H^n$ for $n \geq 30$.
Theorem (Prokhorov-Kovanskij) There do not exist finite-volume Coxeter polyhedra in $H^n$ when $n \geq 996$.
So unless there's a subtlety I'm missing, it seems that my first answer is wrong by the second theorem, and my second answer is doubly-wrong by the first and second theorems.
I can't seem to find the answer in the course notes, and Google isn't helping much either. Is there some obvious description of a lattice in $\operatorname{Isom}(H^n)$ for any $n \geq 2$? Thanks.
I don't know about "obvious", but see Section 6.4 of Dave Witte's book.