We consider finite sums of counting functions on the free group F n and the free monoid M n for n ≥ 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.

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Abstract. We prove that for any prime p ≥ 3 the minimal exponential growth rate of the Baumslag-Solitar group BS(1, p) and the lamplighter group Lp = (Z/pZ) ≀ Z are equal. We also show that for p = 2 this claim is not true and the growth rate of BS(1, 2) is equal to the positive root of x 3 −x 2 −2, whilst the one of the lamplighter group L 2 is equal to the golden ratio (1 + √ 5)/2. The latter value also serves to show that the lower bound of A.Mann from [8] for the growth rates of non-semidirect HNN extensions is optimal.

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