On groups with cubic polynomial conditions

Abstract

Let F<inf>d</inf> be the free group of rank d, freely generated by {y1,. . .,yd}, and let DFd be the group ring over an integral domain D. Given a subset E<inf>d</inf> of F<inf>d</inf> containing the generating set, assign to each s in E<inf>d</inf> a monic polynomial ps(x)=xn+cs,n-1xn-1+. . .+cs,1x+cs,0∈D[x] and define the quotient ringA(d,n,Ed)=DFd〈ps(s)|s∈Ed〉ideal. When ps(s) is cubic for all s, we construct a finite set E<inf>d</inf> such that A(d,n,Ed) has finite rank over an extension of D by inverses of some of the coefficients of the polynomials. When the polynomials are all equal to (x-1)³ and D=Z[16], we construct a finite subset P<inf>d</inf> of F<inf>d</inf> such that the quotient ring A(d,3,Pd) has finite D-rank and its augmentation ideal is nilpotent. The set P<inf>2</inf> is {y1,y2,y1y2,y1-1y2,y12y2,y1y22,[y1,y2]} and we prove that (x-1)3=0 is satisfied by all elements in the image of F<inf>2</inf> in A(2,3,Pd).