From GAP-Forum-Sender@Math.RWTH-Aachen.DE Thu Jun 13 12:17 EDT 1996
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X-Miles: GAP Forum article 930 accepted at 13 Jun 96 17:21 +0100
Date: Thu, 13 Jun 96 17:18:46 +0200
From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de>
To: Multiple recipients of list <GAP-Forum@Math.RWTH-Aachen.DE>
Subject: Re: collection process for free groups
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Dear Forum:

Tony Gaglione and David Joyner asked on June 5:

Does there exist GAP code to implement the collection process for
free groups?

Since nobody has answered so far, others seems to be as puzzled as I
am, what is meant.

GAP can work with free groups and freely reduce products:

gap> f := FreeGroup(3);
Group( f.1, f.2, f.3 )
gap> w1 := f.1 * f.2 * f.2 * f.3^-2 * f.2^2;
f.1*f.2^2*f.3^-2*f.2^2
gap> w2 := f.2^-2 * f.3 * f.2 * f.1;
f.2^-2*f.3*f.2*f.1
gap> w1 * w2;
f.1*f.2^2*f.3^-1*f.2*f.1

Surely you mean something better than that, but can you please explain
what?

Dear Joachim Neubueser and the Gap Forum:

Bill Bogley has the idea of what we are talking about. To be more
specific, we mean the following: Given a finitely generated free group
F and a fixed sequence c1, c2, ... of basic commutators (as in M Hall,
chapter 11), can GAP reduce an arbitrary non-trivial element in F to its
normal form in terms of these basic commutators modulo F_n, where
F_n is the nth term of the lower central series of F?
For details, see for example "The commutator collection process", by
A. Gaglione and H. Waldinger in AMS Contemp Math, vol 109, "Combinatorial
Group Theory". 1990, ed by Fine, Gaglione, Tang. A more algorithmic
description is contained in a paper called "Collection" by G. Havas
and T. Nicholson, written sometime in the 1970's. Sorry, our xerox copy
is so poor that the place it appeared can't be read (Proc of ... ACM
Symposium?).

- David Joyner and Tony Gaglione

Kind regards Joachim Neubueser