Pseudocharacters and the Problem of Expressibility for Some Groups

For a class of extensions of free products of groups, a description is given of the space of the real-valued functions defined on the group G and satisfying the conditions (1) the set { (xy) − (x) − (y) x, y G} is bounded; and (2) (xn) = n (x) for any x G and any n (the set of integers). Let G be an arbitrary group and let S be its subset such that S− 1 = S. Suppose gr(S) is the subgroup of G generated by S. Denote by lS(g) the length of an element g gr(S) relative to the set S. Let V be a finite subset of a free group F of countable rank, and let the verbal subgroup V(F) be a proper subgroup of F. For arbitrary group G, denote by (G) the set of values in the group G of all the words from the set V. This paper establishes the infinity of the set {lS(g): g V(G)}, where G belongs to a class of extension of free products of groups, S = (G) (G)− 1.