On the decidability of the zero divisor problem

Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.