Stable Images of Finite-Index Self-Embeddings of Finite-Lamp Wreath Products

Let F be a nontrivial finite group and let G be the wreath product of F with the integers. We prove that for every injective endomorphism of G with finite-index image, the stable image is infinite. Consequently, no finite-lamp wreath product of this form is strongly scale-invariant through iterations of a single finite-index endomorphism. The proof combines universal slope rigidity, the finite abelian case, finite co-Hopfian rigidity for centerless lamp groups, a periodic-axis argument, and a central lifting induction. This record contains the English and Russian versions of the same preprint.