> < ^ Date: Tue, 03 Apr 2001 09:30:01 +0100 (BST)
> < ^ From: Derek Holt <dfh@maths.warwick.ac.uk >
< ^ Subject: Re: second homology

Dear GAP Forum

Marian Anton wrote:

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?

I am afraid that I cannot help you there. I do not know of any general
algorithmic methods for computing presentations of infinite insoluble
matrix groups, and I am doubtful whether there is anything in GAP that
would be very helpful in that respect. It would certainly be interesting
to try to develop such algorithms.

Have you tried the literature for presentations? I know that
presentations are known for some arithmetic groups, like the Bianchi
groups.

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

Given a finite presentation of G=SL(2,A), it is not difficult to compute
a presentation for a covering group of G, which is a group E, having H
in its centre with E/H ~= G. I even have a GAP procedure to do that.

However, it is not immediately obious how you determine the structure of
H as abelian group from this presention, because this requires finding
a presentation of H which has infinite index in E. There may be some
possibility of doing that however if the group E happens to be automatic.

There is also an algorithm by J.R.J. groves for computing integral homology
groups (J. Algebra 194, 331-361 (1997)) which is implemented in the Magnus
package developed in New York. But this requires a finite complete
rewriting system for G, and I am not too hopeful that your G would have
such a system. But again, you would need a presentation of G to begin
with.

If you do succeed in finding or computing a finite presentation of G,
then I would be interested to hear about it! I might be able to help in
computing H from the presentation.

Derek Holt.

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