We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$\boldsymbol {k}$-SOLVABLE TOPOLOGY ON A FREE GROUP
We prove that, given a finitely generated subgroup H of a free group F, the following questions are decidable: is H closed (dense) in F for the pro-(met)abelian topology? Is the closure of H in F for 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 whether H is 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.
