On endomorphisms of the direct product of two free groups
Abstract
We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated, and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.
Funding source: Fundação para a Ciência e a Tecnologia
Award Identifier / Grant number: SFRH/BD/145313/2019
Funding statement: The author was supported by the grant SFRH/BD/145313/2019 funded by Fundação para a Ciência e a Tecnologia (FCT).
Acknowledgements
The author is grateful to Pedro Silva for fruitful discussions of these topics and to the anonymous referee for helpful comments and suggestions that greatly improved the paper.
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Communicated by: Alexander Olshanskii
References
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Curvature distribution and hyperbolicity
- On Whitehead’s cut vertex lemma
- Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
- On endomorphisms of the direct product of two free groups
-
The
- A note on torsion length and torsion subgroups
- On the separability of subgroups of nilpotent groups by root classes of groups
- The Addition Theorem for two-step nilpotent torsion groups
- A bound for the orders of centralizers of irreducible subgroups of algebraic groups
- The average character degree of finite groups and Gluck’s conjecture
- Character degrees of normally monomial 𝑝-groups of maximal class
- Finite groups with an affine map of large order
- Classification of non-solvable groups whose power graph is a cograph
Artikel in diesem Heft
- Frontmatter
- Curvature distribution and hyperbolicity
- On Whitehead’s cut vertex lemma
- Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
- On endomorphisms of the direct product of two free groups
-
The
- A note on torsion length and torsion subgroups
- On the separability of subgroups of nilpotent groups by root classes of groups
- The Addition Theorem for two-step nilpotent torsion groups
- A bound for the orders of centralizers of irreducible subgroups of algebraic groups
- The average character degree of finite groups and Gluck’s conjecture
- Character degrees of normally monomial 𝑝-groups of maximal class
- Finite groups with an affine map of large order
- Classification of non-solvable groups whose power graph is a cograph
