Erschler and Karlsson in Annales de l'Institut Fourier 60(6):2095-2113, 2010 construct a homomorphism of a finitely generated group G to using a random walk approach. Central to their construction were the word length and a well behaved measure on G. We consider a modified version of this construction using instead of a quasimorphism f of G. Moreover, if a group H acts on G via group automorphisms we show that this technique can be adapted to construct a homomorphism of the semidirect product to , in analogy with the word length case (Bettencourt and Mendes, Appl. Math. Inf. Sci. 9(6):1-7, 2015).

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Random walks on groups,Homomorphisms,Quasimorphisms,Semidirect product

• Mathematics - Natural Sciences