From: Robin Chapman
Subject: Re: faithful representations
Date: Mon, 13 Dec 1999 08:57:07 GMT
Newsgroups: sci.math.research
Keywords: Every group has two- (or finite-)dimensional represetation? (no)
In article ,
Cris Moore wrote:
>
> Is it true that any finitely presented group has a two-dimensional unitary
> representation which is faithful? (e.g. the free group on two generators
> has lots of representations generated by two irrational rotations around
> two different axes.)
No. There are finite groups without faithful 2-dimensional
representation, for example A_5.
> If not, is there always a finite-dimensional one?
A good question. I don't know the answer to this one ...
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
Greg Egan, _Teranesia_
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From: mikeat1140@aol.com (MikeAt1140)
Subject: Re: faithful representations
Date: 13 Dec 1999 01:37:08 GMT
Newsgroups: sci.math.research
By a Theorem of Lipton and Zalcstein
JACM Vol 24 No.3 (1977) pp. 522-526
The word problem for a finitely generated
linear group over a field of characteristic
zero is solvale in logspace.
There are finitely presented groups with
unsolvable word problem.
Your question is:
Is it true that any finitely presented group has a two-dimensional unitary
representation which is faithful?
Thus the answer is 'no'.
However there are many classes of groups for which the answer is 'yes'..
Problem: Is the conjugacy problem for a finitely presented group which have a
two-dimensional unitary
representation solvable by a quantum computer in polynomial time?
**********************************************
Professor Michael Anshel
Department of Computer Sciences R8/206
The City College of New York
New York,New York 10031
==============================================================================
From: Torsten Ekedahl
Subject: Re: faithful representations
Date: 13 Dec 1999 05:47:17 +0100
Newsgroups: sci.math.research
Cris Moore writes:
> Is it true that any finitely presented group has a two-dimensional unitary
> representation which is faithful? (e.g. the free group on two generators
> has lots of representations generated by two irrational rotations around
> two different axes.) If not, is there always a finite-dimensional one?
No, such a group would be residually finite: As it is finiteily
generated the matrix coefficients of its elements will generate a
finitely generated ring so that the group would be a subgroup of
GL_2(R), where R is a finitely generated ring. By looking at quotients
GL_2(R) -> GL_2(R/m) where m runs over the maximal ideals of R (or
modulo such powers if R hadn't been reduced) one gets that GL_2(R) is
residually finite. There are examples of finitely presented groups
without (non-trivial) finite quotients and such a group have no
non-trivial finite dimensional representations by the same argument.
==============================================================================
From: Roger Alperin
Subject: Re: faithful representations
Date: Tue, 21 Dec 1999 13:06:04 -0800
Newsgroups: sci.math.research
Cris Moore wrote:
>
No, not every fp group has a representation by matrices. For example, if
a fg group has a faithful representation by matrices then it has a
solvable word problem, it is residually finite, it is virtually
torsion-free. These are properties that all fp groups do not have.
Roger
> Is it true that any finitely presented group has a two-dimensional unitary
> representation which is faithful? (e.g. the free group on two generators
> has lots of representations generated by two irrational rotations around
> two different axes.) If not, is there always a finite-dimensional one?
>
> This is relevant to a little question in quantum computation.
>
> - Cris Moore, moore@santafe.edu
>
> Mata-Me O Destino Troca-Me O Corpo Muda-Me O Lugar --- Miragaia
> -----------------------------------------------------------------------------
> Cris Moore Santa Fe Institute moore@santafe.edu http://www.santafe.edu/~moore
==============================================================================
From: alperin@my-deja.com
Subject: Re: faithful representations
Date: Thu, 23 Dec 1999 00:59:18 GMT
Newsgroups: sci.math.research
My first guess would be:
given a faithful rep of G in SL_2, it embeds in
SU_2 iff all its solvable subgroups are abelian.
Roger
In article <3853efe3_2@bingnews.binghamton.edu>,
Bob Riley wrote:
> I have had a loooong standing question: which finitely presented
> subgroups of SL(2,C) do/do not imbed in SU(2)? I even asked Serre.
> He didn't know.
>
> R^2
>
>
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From: Joerg Winkelmann
Subject: Re: faithful representations
Date: 2 Jan 2000 13:24:08 -0600
Newsgroups: sci.math.research
Cris Moore wrote:
> Is it true that any finitely presented group has a two-dimensional unitary
> representation which is faithful? (e.g. the free group on two generators
> has lots of representations generated by two irrational rotations around
> two different axes.)
No. In fact the existence of a low-dimensional faithful representation
is a strong condition. Free groups are very special.
For instance, SL(n,Z) for n>=3 is a lattice in SL(n,R) and is
finitely presentable, but by Mostow Margulis Rigidity every representation
of SL(n,Z) on a vector space of dimensional less than n has
a kernel of finite index in SL(n,Z). In particular, it cannot be
faithful.
> If not, is there always a finite-dimensional one?
>
That is a more reasonable question, but the answer is still negative.
By a theorem of Malcev every finitely generated group with
faithful finite-dimensional representation ( over C )
must be residually finite.
But there are finitely presentable groups which are not
residually finite and therefore can not have a faithful
representation.
The first such example is due to Higman, I believe.
There are also some famous finitely presented groups for which
it is not yet known whether they do admit a faithful
representation (fin.dim, over C), e.g. the braid groups B_n.
Recently a colleague of mine in Basle, Dan Krammer, proved that
B_4 admits a faithful representation, but for n>4 this is still
an unsolved problem.
Regards
Joerg
--
jwinkel@member.ams.org