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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 groupH = 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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