On self-similarity of wreath products of abelian groups

Abstract

We prove that in a self-similar wreath product of abelian groups G = BwrX, if X is torsion-free then B is torsion of finite exponent. Therefore, in particular, the group ZwrZ cannot be self-similar Furthemore, we prove that if L is a self-similar abelian group then L ω wrC 2 is also self-similar. We thank the referee for suggestions which improved the original text.

Self-similar products of groups

“…Let us finally give in what appears to be a more natural setting the main result of [5]. Recall that for H ď G we denote by HzG the set of left cosets, namely the set of Hg with g P G.…”

“…In particular, all 2-step finitely generated torsion-free nilpotent groups are self-similar [2]; but lattices in Dyer's example [7] of a Lie group without dilations cannot be self-similar. Free groups are self-similar [11], but Z ≀ Z is not [5,Theorem 1].…”

Self-similar groups of type $$FP_n$$

“…We show the existence of δ in ZQ such that δB is of finite index in B and such that the map f (δb) = b for all b in B together with f | Q = id Q defines a simple virtual endomorphism f : H → G of G. With this, we have the extra nice property that f is an isomorphism between the subgroup H of finite index in G and G itself. Note that by the main result of Dantas and Sidki [19], Theorem C does not hold for metabelian groups, where the Krull dimension of B is bigger than 1.…”

“…It is worth noting that not every finitely generated metabelian group is transitive self-similar. In [19] Dantas and Sidki showed that Z ≀ Z is not a transitive self-similar group. In Section 5 we make explicit calculations about the states of some generating elements of a metabelian self-similar group that has finite index in P U (2, A) from Theorem A.…”

On self-similar Lie algebras and virtual endomorphisms

“…There is an extensive theory of self-similar groups. A link between virtual endomorphisms and self-similar groups was established by Nekrashevych, Sidki in [35], [36] and was later used by Berlatto, Dantas, Kochloukova, Sidki to construct new classes of self-similar groups [2], [17], [18], [26]. As in the theory of self-similar groups we develop a theory of virtual endomorphisms of Lie algebras with applications to self-similar Lie algebras.…”