From: "A. Caranti" <caranti@science.unitn.it>
Subject: Re: Infinite groups
Date: Sat, 19 Feb 2000 20:47:03 +0100
Newsgroups: sci.math.research
Summary: [missing]

On  Wed,  16 Feb  2000  17:59:55 GMT  Thomas  Mattman  posted on  this
mewsgroup  a   question  concerning  the   (in)finiteness  of  certain
groups. I asked Mike Newman, who  has been visiting my Dept, who might
be able to answer this question,  and he pointed me to Rick Thomas.  I
sent  a copy  of  the original  message  to Rick  Thomas, who  replied
privately to Thomas  Mattman. With Rick Thomas's permission,  I am now
posting his message, since there may be other people interested in it.

Andreas Caranti

---------- Forwarded message ----------
Date: Fri, 18 Feb 2000 09:00:20 +0000
From: Rick Thomas
To: Thomas Mattman
Cc: caranti@science.unitn.it
Subject: Infinite groups

Dear Professor Mattman,

Professor Caranti passed on to me your enquiry about the groups

  (2,l,m;q) = <R,S | R^l = S^m = (RS)^2 = (R^2S^2)^q = 1>.

I tend to think of these in the form

  (2,l,m;q) = <a, b | a^2 = b^l = (ab)^m = [a, b]^q = 1>

where a = RS and b = R. The finiteness/infiniteness of
these groups has been completely determined except in
the case (2,3,13;4). Indeed, if we consider the wider
class of groups defined by the presentations,

  (k,l,m;q) = <a, b | a^k = b^l = (ab)^m = [a, b]^q = 1>

there are only five further undetermined cases, namely

  (3,4,9;2)  (3,4,11;2)  (3,4,13;2)  (3,5,6;2)  (3,5,7;2).

I append a list of references at the end which, between them,
contain a proof of this fact (and the enumeration as to
which groups are finite).

I hope this is of some help, but please get back to me if
there are any more details you require.

Best wishes,

Rick Thomas.


[Cox57] H. S. M. Coxeter,
Groups generated by unitary reflections of period two,
Canadian J. Math. 9 (1957), 243--272.

[Edj92] M. Edjvet,
An example of an infinite group,
in W. J. Harvey and C. Maclachlan (eds.),
Discrete Groups and Geometry
(London Math. Soc. Lecture Note Series 173,
Cambridge University Press, Cambridge, 1992),
66--74.

[Edj94] M. Edjvet,
On certain quotients of the triangle groups,
J. Algebra 169 (1994), 367--391.

[EdH96] M. Edjvet and J. Howie,
On the abstract groups (3,n,p;2),
J. London Math. Soc. 53 (1996), 271--288.

[HoP92] D. F. Holt and W. Plesken,
A cohomological criterion for a finitely
presented group to be infinite,
J. London Math. Soc. 45 (1992), 469--480.

[HoT93] J. Howie and R. M. Thomas,
The groups (2, 3, p; q); asphericity and a
conjecture of Coxeter,
J. Algebra 154 (1993), 289--309.


______________________________________________________

  Prof R. M. Thomas,
  Department of Mathematics and Computer Science,
  University of Leicester,
  University Road,          TEL : (+44) 116 2523411
  Leicester LE1 7RH, U.K.   FAX : (+44) 116 2523604

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