Relations between counting functions on free groups and free monoids

TL;DR

This paper investigates the linear relations among sums of counting functions on free groups and monoids, providing a basis and an algorithm to determine equivalence, with applications to bounded cohomology.

Contribution

It characterizes all linear relations between counting functions, constructs an explicit basis, and introduces an algorithm for equivalence testing, advancing understanding of these functions in free groups.

Findings

Complete set of linear relations identified

Explicit basis for the vector space constructed

Algorithm for equivalence determination developed

Abstract

We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n \geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.