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- Title
THE PRO- $\boldsymbol {k}$ -SOLVABLE TOPOLOGY ON A FREE GROUP.
- Authors
MARION, CLAUDE; SILVA, PEDRO V.; TRACEY, GARETH
- Abstract
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.
- Subjects
FREE groups; SOLVABLE groups; FINITE groups; ABELIAN groups; TOPOLOGY
- Publication
Journal of the Australian Mathematical Society, 2024, Vol 116, Issue 3, p363
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788723000162