Virtual retraction properties in groups
Virtual retraction properties in groups
If G is a group, a virtual retract of G is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not.
The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.
Virtual retractions, virtual free factors, property (LR), property (VRC), M. Hall's property
13434–13477
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
1 September 2021
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Minasyan, Ashot
(2021)
Virtual retraction properties in groups.
International Mathematics Research Notices, 2021 (17), .
(doi:10.1093/imrn/rnz249).
Abstract
If G is a group, a virtual retract of G is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not.
The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.
Text
virt_retr-9
- Accepted Manuscript
More information
Accepted/In Press date: 26 August 2019
e-pub ahead of print date: 22 November 2019
Published date: 1 September 2021
Keywords:
Virtual retractions, virtual free factors, property (LR), property (VRC), M. Hall's property
Identifiers
Local EPrints ID: 433889
URI: http://eprints.soton.ac.uk/id/eprint/433889
PURE UUID: f34fdba8-47e1-4c93-9695-ced1681c865c
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Date deposited: 05 Sep 2019 16:30
Last modified: 16 Mar 2024 08:10
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