Groups with semigroup identities: On a conjecture of M. V. Sapir (Q1192777)

Summary: M. V. Sapir conjectured the following: let \(S\) be a finitely generated semigroup satisfying the identity \(x^ 2 = 0\), and assume that \(S\) is a homomorphic image of a subsemigroup of a nilpotent group \(G\). Then \(S\) is finite. If that is true, then a soluble group with a finite basis for its semigroup identities is either abelian or of finite exponent. In this paper we prove the conjecture if either \(\gamma_ 3(G)\) is periodic or \(S\) is 3-generator and \(\gamma_ 4(G)\) is periodic.