# Description of quasimorphisms of the free group

Let $$F$$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $$Q = Q^0_h(F, \mathbb{R})$$ be the space of real homogenous quasimorphisms that vanish on the basis of $$F$$. In Grigorchuk's "Some results on bounded cohomology", he gives a description of $$Q$$ as follows.

Let the length of a conjugacy class of $$F$$ be the word-length of a shortest representative. Then there exists a set $$P$$ of shortest representatives of conjugacy classes of simple elements (i.e., not proper powers) of length at least 2, such that $$P = P^{-1}$$ and every element of $$P$$ is a non-overlapping word: that is, a word $$W$$ that cannot be written as a reduced word $$W = xyx$$. Let $$P^+$$ be a subset containing only one of $$W, W^{-1}$$ for each $$W \in P$$. Let $$e_W$$ be the Brooks quasimorphism corresponding to $$W$$. Then any element of $$Q$$ can be written uniquely as: $$\sum\limits_{W \in P^+} \alpha_W e_W.$$ (Here we are working with pointwise convergence).

Now, in the same article, the author ends by saying that it is a question of some interest to understand which sequences $$(\alpha_W)_{W \in P^+}$$ give rise to elements of $$Q$$. He proves that all $$\ell^1$$ sequences do, but gives an example showing that this is not a necessary condition.

My question is: has there been any further work on this? Is there a complete characterization of which sequences work, or just any further result in this direction?

• I can't remember: does Grigorchuk's article mention the work of Faiziev? Sep 16, 2019 at 15:49
• @YemonChoi The 1993 paper "Pseudocharacters on frre groups" is cited in the paper, but not relating to this question Sep 16, 2019 at 18:32