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Relations between counting functions on free groups and free monoids. (English) Zbl 1451.20005
Summary: 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.

MSC:
20E05 Free nonabelian groups
20M05 Free semigroups, generators and relations, word problems
20J06 Cohomology of groups
05A15 Exact enumeration problems, generating functions
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F28 Automorphism groups of groups
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