Self-similar groups in the sense of an iterated function system and their properties (Q2258493)

A compact topological group \(G\) with a translation-invariant metric \(d\) is called strong self-similar, if it has a proper subgroup \(H\) of finite index allowing a group isomorphism \(\varphi: G \mapsto H\), which is a contraction with respect to \(d\). The authors investigate properties of these groups. In particular, they prove that any strong self-similar group is profinite, i.e., it is topologically isomorphic to an inverse limit of finite discrete topological groups, and obtain certain sufficient conditions for strong self-similarity of a profinite group.