> < ^ Date: Fri, 30 Mar 2001 11:51:24 -0800 (PST)
^ From: Marian Anton <marianfanton@yahoo.com >
> ^ Subject: second homology

Dear gap-forum

I would like to look at the second homology
of SL(2,A) where A is a ring of S-integers.
The example of interest is A=Z[u,1/5] with u
a primitive 5-th root of unity (Z is the ring
of rational integers).

One way to look at the second homology is to
take a finite presentation R-->F-->SL(2,A)
in which F is free and R is the normal closure
of a set of relators. (So I believe that it
is possible to have a finite set of generators
and o finite set of relators). Then I can use
the Hopf formula for the second homology group

```H = R&[F,F]/[R,F]
```

where "&" means intersection.

Here there are a couple of questions (I don't know
gap and I try to make sense of the manual).

1. Suppose I know a finite set of matrices generating
SL(2,A). How can I find the relators?

2. Suppose I know the generators and the relators.
What
strategy I need to calculate the group H above?

3. Is it any other better way to calculate H using
gap? (via cell-complexes for instance)

Sorry for the trouble, but I really need to see an
example.

Comment: I will be also happy if for a given prime p
I can show that the p-torsion of H is zero.

Best regards,
M Anton

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