Everyone knows that it's difficult to compute a general fundamental domain for an arithmetic group but are there any specific examples where such domains have been calculated? I'm mostly interested in results related to orthogonal groups of lattices of type O(2, n).


I've only really been able to track down two such results:


Vinberg's algorithm computes a fundamental domain for reflective groups of type O(1,n); and Gottschling was able to compute a Minkowski reduced domain for the integral symplectic group Sp(4, Z), which can be considered of type O(2, 3).


Would the situation improve if one were to look at a restricted problem, and study fundamental domains of groups generated by p-torsion, for example?



Examples of fundamental domains
Here are some more examples of fundamental domains for arithmetic
groups, although not of the type $O(2,n)$ you are most interested in. 


A situation in which quite something is known on fundamental domains
are the Bianchi groups, i.e., the groups
$PSL\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{-d})})$ for $d$ a positive
square-free integer. These arithmetic groups act on hyperbolic space
$\mathbb{H}^3$ (and therefore can be considered lattices in
$SO^+(1,n)$). The first examples of fundamental domains were computed
by Bianchi in 1892, whence the name. The subsequently developed
technique used to compute fundamental domains in 
the cases of the Bianchi groups was mostly to provide an equivariant
retract of hyperbolic 3-space to a 2-dimensional subcomplex on which
the group acts cocompactly:


* D. Flöge: Zur Struktur der $PSL\_2$
über einigen imaginär-quadratischen Zahlringen. Math. Z. 183 (1983)
255-279
* E.R. Mendoza: Cohomology of $PGL\_2$ over imaginary
quadratic integers. Bonner Mathematische Schriften, 128.


Computations of fundamental domains for Bianchi groups were also the basis for
group cohomology computations of Bianchi groups done in 


* J. Schwermer and K. Vogtmann: The integral homology of $SL\_2$ and $PSL\_2$ of
euclidean imaginary quadratic integers. Comment. Math. Helv. 58 (1983)
573-598.


Several algorithms have been described for computing the fundamental
domain for Bianchi groups. You can have a look at [the thesis of A. Rahm](http://hal.archives-ouvertes.fr/docs/00/58/27/70/PDF/these.pdf)
in which an algorithm is described which is also implemented using
Pari/GP. This allows to do a couple of computations of fundamental
domains for Bianchi groups:


* A. Rahm and M. Fuchs. JPAA 215 (2011) 1443-1472
* A. Rahm. LMS Journal of Computation and Mathematics 16
(2013), 344-365.


Note that only finitely many of the Bianchi groups are generated by
reflections (M. Belolipetsky and J. McLeod: Reflective and
quasi-reflective Bianchi groups. arXiv:1210.2759), and so Vinberg's
algorithm does not apply to most Bianchi groups. 


The technique of retraction onto a nice subcomplex on which the group
acts cocompactly has also been used by Soulé (C. Soulé: The cohomology
of $SL\_3(\mathbb{Z})$. Topology 17 (1978) 1-22). This technique has
subsequently been improved so that today it is possible to compute
fundamental domains for $SL\_n(\mathbb{Z})$ for $n\leq 8$, see the
works of Elbaz-Vincent, Gangl, Soulé and others.


For the symplectic group $Sp\_4(\mathbb{Z})$, you may also be
interested in the paper A. Brownstein and R. Lee: Cohomology of the
symplectic group $Sp\_4(\mathbb{Z})$ I: the odd torsion
case. Transactions AMS 334 (1992) 575-596. They do not compute a
fundamental domain, but the relation between the
$Sp\_4(\mathbb{Z})$-quotient and the moduli space of genus $2$ surfaces
might be interesting anyway. 
In a similar direction, for Hilbert modular groups, i.e.,
$SL\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{d})})$ for $d$ a positive
square-free integer, a lot is known about the quotients
$SL\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{d})})\backslash\mathbb{H}^2\times\mathbb{H}^2$
because these quotients have the structure of complex
surfaces. There are several papers of Hirzebruch about this
topic. This does not give precise information about the 
fundamental domain inside $\mathbb{H}^2\times\mathbb{H}^2$, but it
does provide some information on the group action, the quotient, group
cohomology etc. Of course, this can be generalized to other types of
groups, but I know nothing about Shimura-varieties, so I cannot say
anything about that... 


