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.