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_ Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== 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 > > Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== 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