SL2 over group rings of cyclic groups

Abstract

Let R denote the group ring k C of an infinite cyclic group C = 〈 σ〉 over a field k which is commutative or non-commutative. Define R o to be the subring k 〈σ 2〉, and let s=( σ 0 0 σ −1 ) be an element of GL 2( R). A natural explicit presentation is given for E 2( R o ), the group of elementary transformations in GL 2( R o ), in terms of the generating subgroups SL 2( k ) and SL 2( k ) s, thus generalizing a theorem of Ihara. For k a finite field and for a positive integer n, the further condition s 2 n = 1 produces SL 2(k〈τ〉) 〈( τ n 0 0 τ −n )〉 where τ has order 2 n. With this, the problem of the structure and finiteness of certain classes of groups related to SL 2(2 m ) is settled.