From: serge bouc
Subject: Re: subgroups of SL_2(R)
Date: 19 Sep 2000 12:00:03 -0500
Newsgroups: sci.math.research
Summary: [missing]
david carlton wrote:
> Does anybody know of results along the following lines?
>
> Let R be a commutative ring, and define u(x) to be the matrix
>
> 1 x
> 0 1
>
> Also, let M be a subgroup of the additive group of R such that the
> smallest subring of R containing M is R. Finally, let G be a subgroup
> of SL_2(R) that contains the image of SL_2(integers) and u(M). Then:
>
> Does G = SL_2(R)?
>
> I can show that the answer is yes in some cases, e.g. when R is local
> and M = R. The proof in that case isn't anything clever: I just
> conjugate elements of u(M) by appropriate elements of SL_2(integers)
> and multiply them together for a while until I get everything. I have
> no idea if the answer is yes in general, but I'd like to hear about
> other cases where this is known (or, for that matter,
> counterexamples).
>
> thanks,
> david carlton
> carlton@math.stanford.edu
The following theorem of Dickson has the same flavour :
Let p be an odd prime number, and F=GF(p). Let x be a non zero
element of an algebraic closure of F, and set F'=F(x).
Then the subgroup of SL_2(F') generated by the elements
1 0
1 1
and
1 x
0 1
is equal to SL_2(F'), except if p=3 and x^2=-1.
--
Serge Bouc
sbouc@nnx.com
bouc@math.jussieu.fr
www.math.jussieu.fr/~bouc
==============================================================================
From: Roger Alperin
Subject: Re: subgroups of SL_2(R)
Date: 19 Sep 2000 12:00:10 -0500
Newsgroups: sci.math.research
I believe it is an open question as to whether or not SL_2(R)=E_2(R) for
the ring R=Z[t,1/t].
The elementary group E_2(R) certainly contains SL_2(Z) and the upper
triangular group M=R.
Can you prove the similar result in this case of R.
Roger
==============================================================================
From: Torsten Ekedahl
Subject: Re: subgroups of SL_2(R)
Date: 21 Sep 2000 07:00:02 -0500
Newsgroups: sci.math.research
david carlton writes:
> Does anybody know of results along the following lines?
>
> Let R be a commutative ring, and define u(x) to be the matrix
>
> 1 x
> 0 1
>
> Also, let M be a subgroup of the additive group of R such that the
> smallest subring of R containing M is R. Finally, let G be a subgroup
> of SL_2(R) that contains the image of SL_2(integers) and u(M). Then:
>
> Does G = SL_2(R)?
>
> I can show that the answer is yes in some cases, e.g. when R is local
> and M = R. The proof in that case isn't anything clever: I just
> conjugate elements of u(M) by appropriate elements of SL_2(integers)
> and multiply them together for a while until I get everything. I have
> no idea if the answer is yes in general, but I'd like to hear about
> other cases where this is known (or, for that matter,
> counterexamples).
In general the answer is no. A reference for that is Bass' big book
named 'Algebraic K-theory', let me just give a short summary: One
defines SK_1(R) for R a commutative ring as the limit when n goes to
infinity of SL_n(R)/E_n(R), where E_n(R) is the subgroup generated by
elementary matrices. In the present case G is E_2(R) as it is known
that SL_2(Z) + E_2(Z). Now (proof in loc. cit),
1) SL_2(R) -> SK_1(R) is surjective when R is a Dedeking ring and is
evidently trivial on G.
2) SK_1(R[x,y]/(x^2+y^2-1)), where R is the real numbers is
non-trivial. In fact the proof of non-triviality in loc. cit. gives an
explicit invariant on SL_2 of the ring which is trivial on E_2.