TITLE: Computations in group cohomology
QUESTION [7 upvotes]: Hello,
Given a finitely presentable group $G$, I'm interested in the cup-product from $H^1$ to $H^2$ with real coefficients. I want to know if this is explicitly computable (with a computer) with a presentation of the group.
More precisely, I want a program that takes the generators and relations as entries and returns the dimension of the $H^1$ and a finite generating set of linear relations between the cup-products of every couple of elements in a basis of $H^1$. (I am not really interested in all the $H^2$)
Does this seem possible ?
I precise that I am not really familiar with group cohomology and I ask this question because it is certainly known if such a problem cannot be resolved with an efficient algorithm.
The problem comes in the study of Kähler groups where this cup-product plays an important role.
Thank you.

REPLY [3 votes]: Using Andy's comment and Theorem 6.1 in this paper you can easily work out a computer program to calculate what you wish from a presentation. You will only need to work with groups of nilpotency class $2$.