I did manage to get an answer -

First, using LowIndexSubgroupsFpGroup, then using the same again on one of the
subgroups found, I found an subgroup of index 144 whose AbelianInvariants include
0.

This approach was suggested by someone on the list [ apologies to that person, I
can't recall who ]

Later, someone (not on the GAP list) pointed out a theorem that gives the answer in
a non-group-theoretic way (the group has strong links to geometry).

The group is a quotient of the Coxeter group of type 4,3,3,5. Therefore, the
presentation is:

F := FreeGroup( "a", "b", "c", "d", "e");;
a := F.1;; b := F.2;; c := F.3;; d := F.4;; e := F.5;;
G := F / [
a^2,
b^2,
c^2,
d^2,
e^2,
(a*b)^4,
(a*c)^2,
(a*d)^2,
(a*e)^2,
(b*c)^3,
(b*d)^2,
(b*e)^2,
(c*d)^3,
(c*e)^2,
(d*e)^5,
(b*c*d*e)^15     # Extra relator not found in the Coxeter group presentation
];;

then

maxord := 12;;
lis := LowIndexSubgroupsFpGroup( G, TrivialSubgroup( G ), maxord);;
lis4 := LowIndexSubgroupsFpGroup( lis[4], TrivialSubgroup( G ), maxord);;
ndx4 := List(lis4, x -> Index(G,x));;
abinv4 := List (lis4, x -> AbelianInvariants(x));;
ndx4[10];
lis4[10];
abinv4[10];

proves the group to be infinite.

On 23 Apr 2002 at 13:46, Joachim Neubueser wrote:

Dear Dr. Hartley,

On April 8 you had asked the GAP Forum how to check whether a
particular finitely presented group is finite or infinite. After an
answer from Dr. Stephan Rosebrock who had just pointed out that the
function 'IsFinite' may indeed not give an answer, I had written the
appended letter. This went to the GAP Forum as usual when we answer a
question that has been put to the Forum.

Since then, I have not heard from you either individually nor via the
'gap-trouble' address. However since I am always interested to see in
examples if and how far the functions that we provide for finitely
presented groups do work in 'real' examples, I am curious to know, if
you succeeded to use the suggestions in my letter to decide your
problem. If not, I want to renew my offer to help with the
investigation (without any guarantee that we will be able to answer
it). in any case I would appreciate to see the presentation. As
indicated, since this is in principle an undecidable problem, we are
alsways interested to see practically relevant examples in order to
learn more about the applicability of the existing methods.

Looking forward to hear from you

with kind regards Joachim Neubueser

---------------------------------------------------------------------
Pr9of. em J. Neubueser
Lehrstuhl d fuer Mathematik
RWTH Aachen, Germany.
---------------------------------------------------------------------

----- Forwarded message (env-from jneubues) -----

Dear GAP forum,

Michael Hartley's letter, copied below, has in principle already
correctly been answered by Stephan Rosebrock, namey that infiniteness
of a finitely presented group is an algorithmically undecidable
problem. However for any given finite presentation GAP provides a
variety of methods with which (in a 'trial and error' fashion) one can
try to solve the problem in this particular case. E.g. one may look
for abelian or nilpotent quotients, or for subgroups of finite index
which in turn may have infinite abelian or nilpotent quotients etc.
Since such methods can only be tried individualy 'by hand', they are
not invoked in 'IsFinite'.

I would therefore suggest that you send your presentation to
'gap-trouble' where some members of the GAP support team may have a
look if some such method may answer your particular problem. If it
does, the result should eventually been reported to the Forum, since
the question was asked there, however the rest of the Forum members
should not be bothered with the discussion that may or may not lead to
an answwer.

Kind regards Joachim Neubueser

I have a group, a quotient of a free group, and I want to check
whether or not it is infinite. I am almost certain that it is.

> What is the best way to do that? IsFinite() didn't seem like it was
going anywhere.

Yours, Mike H...

Michael Hartley : Michael.Hartley@sit.edu.my
Dean, School of Multimedia and Engineering,
Sepang Institute of Technology

----- End of forwarded message (env-from jneubues) -----

Michael Hartley : Michael.Hartley@sit.edu.my
Dean, School of Multimedia and Engineering,
Sepang Institute of Technology

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