Relations between counting functions on free groups and free monoids

Abstract

We study counting functions on the free groups $F_n$ and free monoids $M_n$ for $n\ge 2$, which we introduce for combinatorial approach to famous Brooks quasimorphisms on free groups. Two counting functions are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a simple graphical algorithm to determine whether two given counting functions are equivalent. In particular, this yields an algorithm to decide whether two linear combinations of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.