Program for computing group cohomology Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space? 
I mainly care about infinite groups.
 A: For groups given by matrix generators, most questions are undecidable. For groups given by a presentation, many questions are undecidable anyhow:
(un)decidability in matrix groups
Is any interesting question about a group G decidable from a presentation of G?
computing abelianizations
A: Computing group cohomology
One should probably  see this in terms of computing resolutions of a group;  then it is a partly a question of what data defines the group in question, and clearly a computation is not always possible.  
For one approach using Homological Perturbation Theory, see papers involving Larry Lambe at 
http://mssrc.com
for example 
Computing Resolutions Over Finite p-Groups, with Johannes Grabmeier, Algebraic Combinatorics and Applications, A. Betten, A. Kohnert, R. Laue, and A. Wassermann (Eds), Springer-Verlag, Heidelberg, (2001). 
See also papers of Johannes Huebschmann, such as 
MR1031239 (92c:20093)
Huebschmann, Johannes(D-HDBG)
Cohomology of metacyclic groups.
Trans. Amer. Math. Soc. 328 (1991), no. 1, 1–72. 
My paper with Razak Salleh`Free crossed crossed resolutions of
groups and presentations of modules of identities among
relations', LMS J. Comp. and Math.  2 (1999) 28-61. brought forward the idea of computing resolutions not by "killing kernels" but by "constructing a home for a contracting homotopy", i.e. working to construct the universal cover with a contracting homotopy: this was kind of inspired by Homological Perturbation Theory, where the construction of homotopies is paramount.
Carried much further in this  direction and well implemented  is the work of Graham Ellis on Homological Algebra Programming at http://hamilton.nuigalway.ie/  .
A: I'm going to assume you mean a group given by a finite presentation.  The program GAP (see http://www.gap-system.org/) has algorithms to compute $H^1(G;M)$ for a finitely presentable group $G$ and a $G$-vector space $M$.  I also wrote some code at some point for dealing with $M$ a $G$-vector space over a field of finite characteristic (it is a little more efficient than GAP, and can deal with "larger" groups and vector space); the code can be downloaded here, and was used in my paper
A. Putman
The Picard group of the moduli space of curves with level structures,
Duke Math. J. 161 (2012), no. 4, 623–674.
For $H^k(G;M)$ for $k \geq 2$, this is impossible.  In
C. M. Gordon, Some embedding theorems and undecidability questons for
groups, in: Combinatorial and Geometric Group Theory, A. J. Duncan,
N. D. Gilbert, and James Howie, eds., London Math. Soc. Lecture Note
Series 204 (CUP, 1995), 105-110.
it is shown that determining whether $H^2(G;M) = 0$ for a finitely presentable group $G$ is undecidable.
EDIT : Since there has been some discussion in the comments about which classes of finitely presentable groups possess algorithms for computing their homology, I thought I'd point out the paper
Bridson, M and Reeves, L "On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups", http://people.maths.ox.ac.uk/bridson/papers/BReeves/
This paper shows that there exists an algorithm for calculating the homology groups of an automatic group.  This is a fairly broad class of group (eg it includes mapping class groups by a famous theorem of Lee Mosher, though it doesn't include higher rank lattices).  But I don't know how practical the given algorithm is.
