Let Γ be a nonelementary Kleinian group and H<Γ finitely generated, proper subgroup. We prove that if has finite covolume, then the profinite completions of H are not isomorphic. If index in Γ, there is onto which maps but does not. These results streamline existing proofs exist full-sized groups profinitely rigid absolute sense. They build on circle ideas can used to distinguish among subgroup...

The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer [Formula: see text] we prove that if is group the centralizer every nontrivial element has rank at most text], then either pro-[Formula: or Further, not virtually group, text].

We achieve several results. First, we develop a variant of the theory absolute Galois groups in context many sorted structures. Second, provide method for coding structures, so they can be interpreted some monster model with an additional predicate. Third, prove “Weak Independence Theorem” pseudo-algebraically closed (PAC) substructures ambient structure no finite cover property (nfcp) and [For...

Two constructions are described: one gives soluble groups of derived length 4, the other uses acting on a rooted tree.