1 results
THE PRO-
$\boldsymbol {k}$-SOLVABLE TOPOLOGY ON A FREE GROUP
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 116 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 22 December 2023, pp. 363-383
- Print publication:
- June 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
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.
