TITLE: Are finite presentations of arithmetic groups computable?
QUESTION [10 upvotes]: In this famous paper by Borel and Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, it is proved that, in characterisitic zero, arithmetic groups are finitely presented. I have an extremely vague idea of how the argument goes, but I was wondering if anything was known about how this would be done algorithmically. To be specific, (following work by Grunewald and Segal) an arithmetic group $\Gamma \leq \mathrm{GL}(n,\mathbb C)$ is said to be expicitly given if we have the following data:

a system $S$ of polynomial equations defining the algebraic group $V(S)= \mathcal G$ (which contains $\Gamma$), 
a known upper bound  $k$ for the index $[\mathrm{GL}(n,\mathbb Z)\cap \mathcal G:\Gamma]$, and
a procedure which given $g \in  \mathrm{GL}(n,\mathbb Z)\cap \mathcal G$ will decide if $g \in \Gamma$.

Then my question is: If $\Gamma$ is an explicitly given arithmetic group, is it possible to compute the presentation of $\Gamma$ or, even better, to get the orbihedron whose fundamental group is $\Gamma$, which is used in the construction of Borel and Harish-Chandra?
If there is a reference for this, or if this is known to be impossible, that would be awesome!

REPLY [3 votes]: You might want to study the work of Detinko, Flannery, and co-authors. For example:
Detinko, A.; Flannery, D. L.; Hulpke, A., Zariski density and computing in arithmetic groups,  ZBL06825254.
Detinko, A. S.; Flannery, D. L.; Hulpke, A., Algorithms for arithmetic groups with the congruence subgroup property., J. Algebra 421, 234-259 (2015). ZBL1319.20040.
(moving comment up to the main body): They find things like the congruence depth (the index of the principal congruence subgroup),  so the co-volume,  and then (at least in the special linear and symplectic case you can use the Minkowski model (the PSD cone) of the symmetric space to construct the fundamental polyhedron (the bound tells you when to stop), so pretty much what you are asking for. 
In the real hyperbolic case, there is the very interesting paper of Mark and Paupert:
Presentations for cusped arithmetic hyperbolic lattices, by Alice Mark and Julien Paupert.