Invariant functions on free groups and special HNN-extensions

In this paper we are considering questions about the possibility of existence of invariant nontrivial pseudocharacters on free groups. It is proved that nontrivial pseudocharacters exist on a certain type of HNN-expansions in complex cases.
We got some results about the width of verbal subgroups generated by words from commutator subgroup and non-triviality of the second group of bounded cohomologies for considered HNN-expansions. Thus, partial answer to the question, formulated R. I. Grigorchuk, is received.
Pseudocharacter is the real functions $f$ from group $G$ to $\mathcal{R}$ such that $|f(xy) - f(x) - f(y)|\leq \varepsilon$ for some $\varepsilon>0$ and for any $ x,y\in G$ and $ f(x^n) = nf(x)$ $\forall n\in\mathbb{Z}$, $\forall x\in G$. A pseudocharacter is called non-trivial if $\varphi(ab)-\varphi(a)-\varphi(b)\neq 0$ for some $a, b\in G$. Existence of nontrivial pseudocharacters on a group is connected with many important characteristics and properties of groups.
The notion of pseudocharacter was introduced by A. I. Shtern. Sufficient conditions of the existence of nontrivial pseudocharacters for free products with amalgamation and HNN-extensions for which associated subgroups are different from the base group were found by R. I. Grigorchuk and V.G. Bardakov. Nontrivial pseudocharacters exist on groups with one defining relation, and at least three generators.
Questions about conditions of existence of nontrivial pseudocharacters for groups with one defining relation and two generators and for descending HNN-extensions remain open. These questions in many cases are reduced to constructing nontrivial pseudocharacters on free groups, invariant with respect to special type of endomorphisms.
In this paper we prove existence of nontrivial pseudocharacters for free groups $F_n$, $n>1$, which are invariant with respect to certain types of endomorphisms. It is proved that nontrivial pseudocharacters exist on some descending HNN-extensions.
This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik.
Bibliography: 17 titles.