Pseudocharacters and the problem of expressibility for some groups (Q1604362)

A quasicharacter of a semigroup \(S\) is a real-valued function \(f\) on \(S\) such that the set \(\{f(xy)-f(x)-f(y)\mid x,y\in S\}\) is bounded. A pseudocharacter of a semigroup \(S\) is a quasicharacter \(f\) that satisfies \(f(x^n)=nf(x)\) for all \(x\in S\) and all natural numbers \(n\) (all integers if \(S\) is a group). The set \(KX(S)\) of quasicharacters of a semigroup \(S\) is a vector space. The subspace of \(KX(S)\) consisting of all pseudocharacters will be denoted by \(PX(S)\). Let \(H=A*B\) be the free product of two groups \(A\) and \(B\). Let \(T_1\) and \(T_2\) be subgroups of the automorphism groups of \(A\) and \(B\), respectively. Put \(T=T_1\times T_2\). The authors describe the spaces of pseudocharacters of \(H\) and also of \(T\cdot H\). Another major result obtained by the authors is the following: If \(V\) is a finite subset of the free group \(F\) such that the verbal subgroup \(V(F)\) is a proper subgroup of \(F\), then the verbal subgroup \(V(G)\) of \(G\) (where \(G\) is a group) has infinite width.