Geodesic
Pseudocharacters on free groups
Izvestiya. Mathematics , Tome 44 (1995) no. 1, pp. 119-141.

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The perturbations of additive real characters on a free group $F$ are studied. A description is given of the space of its pseudocharacters, i.e., the real functions $f$ on $F$ such that the set $\{f(xy)-f(x)-f(y)$; $x,y\in F\}$ is bounded and $f(x^n)=nf(x)$ $\forall\,n\in\mathbf Z$, $\forall\,x\in F$. 
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V. A. Faiziev. Pseudocharacters on free groups. Izvestiya. Mathematics , Tome 44 (1995) no. 1, pp. 119-141. http://geodesic.mathdoc.fr/item/IM2_1995_44_1_a5/

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