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A group is called free-by-free if it is the semi-direct product of two finitely generated free groups. A group is coherent if any finitely generated subgroup is finitely presented, and incoherent otherwise. In this paper, the authors provide evidence towards the conjecture (due independently to the authors and Dani Wise) that every free-by-free group is incoherent. To do this, they give a homological condition which lets them conclude that the free-by-free group has a finite index subgroup which surjects onto ℤ with finitely generated kernel; standard arguments imply that this kernel cannot be finitely presented. As an important special case, they show that if the free-by-free group is hyperbolic and virtually special, then it is incoherent.
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This content will become publicly available on July 8, 2026
On two‐generator subgroups of mapping torus groups
Abstract We prove that if is the mapping torus group of an injective endomorphism of a free group (of possibly infinite rank), then every two‐generator subgroup of is either free or a (finitary) sub‐mapping torus. As an application we show that if is a fully irreducible atoroidal automorphism, then every two‐generator subgroup of is either free or has finite index in .
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- Award ID(s):
- 1905641
- PAR ID:
- 10613855
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 112
- Issue:
- 1
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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