Nontrivial pseudocharacters on groups with one defining relation and nontrivial centre

The problem concerning existence conditions for nontrivial pseudocharacters on one-relator groups with nontrivial centre is completely solved. It is proved that a nontrivial pseudocharacter exists on a group of this type if and only if the group is nonamenable. A pseudocharacter is a real function on a group for which the set $\{f(xy)-f(x)-f(y);\, x, y\in F\}$ is bounded and $ f( x^n)=nf(x)$ for all $n\in\mathbb{Z}$ and $x\in F$. The existence of pseudocharacters is related to many important characteristics and properties of groups, such as the cohomology groups and the width of verbal subgroups. From our results for pseudocharacters we obtain corollaries concerning the width of verbal subgroups and the second bounded cohomology group for the one-relator groups with nontrivial centre.
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