THE PRO--SOLVABLE TOPOLOGY ON A FREE GROUP

Abstract

AbstractWe prove that, given a finitely generated subgroupHof a free groupF, the following questions are decidable: isHclosed (dense) inFfor the pro-(met)abelian topology? Is the closure ofHinFfor the pro-(met)abelian topology finitely generated? We show also that if the latter question has a positive answer, then we can effectively construct a basis for the closure, and the closure has decidable membership problem in any case. Moreover, it is decidable whetherHis closed for the pro-$\mathbf {V}$topology when$\mathbf {V}$is an equational pseudovariety of finite groups, such as the pseudovariety$\mathbf {S}_k$of all finite solvable groups with derived length$\leq k$. We also connect the pro-abelian topology with the topologies defined by abelian groups of bounded exponent.