Erschler and Karlsson [3] constructed a homomorphism from a finitely generated group to the reals through a random walk approach. In our previous work, similar results were obtained for a semidirect product using the word length [1]. In [2], the word length was replaced by a quasimorphism. It turns out that, working with quasimorphisms made it possible to generalize further the construction to an infinite, finitely generated semigroup S, endowed with a probability measure. Specifically, using semigroup quasimorphisms, we construct a homomorphism from S to (R,+). [1]Bettencourt, G. and Mendes, S., Homomorphisms to R of semidirect products: a dynamical construction, Appl. Math. Inf. Sci., 9.6, 2015, pp. 1--7. [2]Bettencourt, G. and Mendes, S., Homomorphisms to R generated by quasimorphisms, Mediterr. J. Math., 13, 2016, pp. 3205--3219. [3]Erschler, A., and Karlsson, A., Homomorphisms to R constructed from random walks, Annales de l'Institut Fourier, 60.6, 2010, pp. 2095--2113.

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Random walks on semigroups,Semigroup homomorphism,Semigroup quasimorphism

• Mathematics - Natural Sciences