The Mal'tsev-Nielsen equation in a free metabelian group of rank two

Abstract

Let F 2 denote the free group of rank two with free generators a and b. Nielsen [i, 2] and Mal'tsev [3] have proved'that elements x and y of F 2 are its free generators if and only if the disjunction of equations with respect to z ~ = ~ l z [ x , y ] ~ l = [a, b] has a s o l u t i o n , where [u , v] d e n o t e s t h e c o m m u t a t o r o f e l e m e n t s u and v , i . e . , [u , v] = UVU-Iv -1 The aim of the present note is to prove the analogous result for a free metabelian group of rank two. Let F 2 be a free group of rank two with free generators x I and x~ and F~ 2) be its second commutant, i.e., F~ 2) = [F! I), F~I)], where F! l) = [F2, F2]. Then F2/F! ~) is a free metabelian group of rank two with free generators~ ,.x I and x 2 (we do not distinguish between elements xl, x 2 of F 2 and their images in F2/F[2)). In the proofs we will use derivations of the group ring Z [F~/F~ ~] with values in Z [F/F~I']. All the necessary definitions and properties of these derivations are contained in [4, 5] and detailed proofs can be found in [6-8] and [4]. LEMMA i. If elements u and v of the group)F2/F~ 2) satisfy the equation [u, v] = [xl, x2] , t-~en u and v are free generators of F2/F! 2 . Proof. Direct computations in Z [F21 give the following equations: O[xl, x2]/Oxx = t [ x , z2]x2, 0 [z~, x~l/Ox~ = x~-[z~ , x~]. I f w(xz , x 2) i s an e l e m e n t o f F2 and u and r a r e a r b i t r a r y e l e m e n t s o f F2, t h e n t h e f o l l owing r u l e ( t h e c h a i n r u l e ) h o l d s :