“…Furthermore, by taking the group morphism of H 1 into Z ≀ H 1 , we see that the image of h 1 is the generator p1, 0q of the active group, while for every j, the image of h 1 1 j is of the form p0, f j q where f j has finite support. The following result is essentially due to Kaimanovich and Vershik [26, Proposition 6.1], [21,Theorem 1.3], and has been studied in a more general context by Bartholdi and Erschler [6]: Lemma 9.2. Consider the wreath product Z ≀ H 1 where H 1 is not trivial, and let µ be a measure on it such that the projection of µ on Z gives a transient walk and the projection of µ on H 1 Z is finitary and non-trivial.…”