Journal of the Mathematical Society of Japan

Mapping tori with first Betti number at least two


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We show that given a finitely presented group G with β 1 ( G ) 2 which is a mapping torus Γ θ for Γ a finitely generated group and θ an automorphism of Γ then if the Alexander polynomial of G is non-constant, we can take β 1 ( Γ ) to be arbitrarily large. We give a range of applications and examples, such as any group G with β 1 ( G ) 2 that is F n -by- Z for F n the non-abelian free group of rank n is also F m -by- Z for infinitely many m . We also examine 3-manifold groups where we show that a finitely generated subgroup cannot be conjugate to a proper subgroup of itself.

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J. Math. Soc. Japan, Volume 59, Number 2 (2007), 351-370.

First available in Project Euclid: 1 October 2007

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Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57N10: Topology of general 3-manifolds [See also 57Mxx]

mapping torus BNS invariant Alexander polynomial


BUTTON, Jack O. Mapping tori with first Betti number at least two. J. Math. Soc. Japan 59 (2007), no. 2, 351--370. doi:10.2969/jmsj/05920351.

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