# Groups, Geometry, and Dynamics

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**Volume 12, Issue 4, 2018, pp. 1485–1521**

**DOI: 10.4171/GGD/476**

Published online: 2018-11-01

Relations between counting functions on free groups and free monoids

Tobias Hartnick^{[1]}and Alexey Talambutsa

^{[2]}(1) Justus-Liebig Universität Giessen, Germany

(2) Steklov Mathematical Institute of RAS, Moscow, Russia, and National Research University Higher School of Economics, Mosc

We study counting functions on the free groups $F_n$ and free monoids $M_n$ for $n \geq 2$, 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.

*Keywords: *Bounded cohomology, free groups, counting function, counting quasimorphism, Brooks quasimorphism

Hartnick Tobias, Talambutsa Alexey: Relations between counting functions on free groups and free monoids. *Groups Geom. Dyn.* 12 (2018), 1485-1521. doi: 10.4171/GGD/476