Radicals and Plotkin’s problem concerning geometrically equivalent groups

Abstract

If G and X are groups and N is a normal subgroup of X, then the Giclosure of N in X is the normal subgroup X G = T fker 'j' : X ! G; with N µ ker 'g of X. In particular, 1 G = RGX is the Giradical of X. Plotkin (2, 6, 3) calls two groups G and H geometrically equivalent, written G» H, if for any free group F of flnite rank and any normal subgroup N of F the Giclosure and the Hiclosure of N in F are the same. Quasiidentities are formulas of the form ( V in wi = 1! w = 1) for any words w; wi (in) in a free group. Generally geometrically equivalent groups satisfy the same quasiidentiies. Plotkin showed that nilpotent groups G and H satisfy the same quasiidenties if and only if G and H are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups; see the Kourovka Notebook (2). We provide a counterexample.