Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}\_n(\mathbb{Z})$ on $\operatorname{SL}\_n(\mathbb{R})/\operatorname{SO}\_n(\mathbb{R})$ for each $n \leq 6$. In the case where $n = 2$, one obtains the familiar fundamental domain for the action of $\operatorname{SL}\_2(\mathbb{Z})$ on the complex upper half-plane. The case where $n = 3$ is studied in detail in the paper entitled "Hecke Operators and the Fundamental Domain for $\operatorname{SL}(3, \mathbb{Z})$" by Daniel Gordon et al.


Are there analogous computations in the literature of explicit fundamental domains for the action of the orthogonal group $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$, at least for some small values of $n$? I am particularly interested in the case where $n = 2$.


What I know: I understand that computing such fundamental domains is difficult in general. I'm aware of the construction of Borel and Harish-Chandra via Siegel domains, but I'm not sure whether it's possible to make their construction explicit in the way that Minkowski was able to do.



Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$
For information about these groups up through dimension 17, see:


Vinberg, È. B.
The groups of units of certain quadratic forms. (Russian)
Mat. Sb. (N.S.) 87(129) (1972), 18–36.
English translation [Math. USSR-Sb. 87 (1972), 17–35].


Vinberg shows up through dimension 17 that $O(n,1; \mathbb{Z})$ has a finite index subgroup generated by reflections. He gives an explicit polygon for this reflection group, i.e., a fundamental domain for its action. Then $O(n,1; \mathbb{Z})$ is generated by this reflection group along with the symmetry group of the polygon.


He has a later paper with Kaplinskaja that studies 18 and 19:


Vinberg, È. B.; Kaplinskaja, I. M.
The groups O18,1(Z) and O19,1(Z). (Russian)
Dokl. Akad. Nauk SSSR 238 (1978), no. 6, 1273–1275.
English translation: Soviet Math. Dokl. 19 (1978), no. 1, 194–197.


I believe there are some references for (slightly) higher dimensions as well. You might look at some papers of Allcock, for example.


Edit: For the special case $n=2$, it is reflections in the sides of a $(2,4,\infty)$ triangle, i.e., angles $\pi/2$, $\pi/4$, and one ideal vertex. This can be seen directly from the diagram in Table 5 of Vinberg's paper.


