# Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$

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.

• $O(2,1)(\mathbb{Z})$ is commensurable with $\mathrm{PSL}_2(\mathbb{Z})$ and $O(3,1)(\mathbb{Z})$ with $\mathrm{PSL}_2(\mathbb{Z}[i])$, so you should be able to recover fundamental domains for these two from fundamental domains present in the literature. – Aurel Mar 28 at 13:36
• – Aurel Mar 28 at 13:40

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.