Nontrivial Pseudocharacters on Groups and Their Applications

In this paper questions of the existence of non-trivial pseudocharacters for different classes of groups and the most important applications of pseudocharacters are considered. We review the results obtained for non-trivial pseudocharacters of free group constructions. Pseudocharacter is the real functions f on a group G such that 1) |f(ab) − f(a) − f(b)| ≤ ε for some ε > 0 and for any a, b ∈ G and 2) f(xn) = nf(x) ∀n ∈ Z, ∀x ∈ G. Existence of non-trivial pseudocharacters implies the results for second group of bounded cohomologies and the width of verbal subgroups. Results of R.I. Grigorchuk, V.G. Bardakov, V.A. Fayziev and author on this topic are examined. Theorems about conditions of the existence of non-trivial pseudocharacters on such group objects as free products with amalgamation, HNN-extensions, group with one defining relation, anomalous products are given in the article. © 2018, Pleiades Publishing, Ltd.