## Journal of the Mathematical Society of Japan

### Mapping tori with first Betti number at least two

Jack O. BUTTON

#### Abstract

We show that given a finitely presented group $G$ with $\beta_1(G)\geqq 2$ which is a mapping torus $\Gamma_\theta$ for $\Gamma$ a finitely generated group and $\theta$ an automorphism of $\Gamma$ then if the Alexander polynomial of $G$ is non-constant, we can take $\beta_1(\Gamma)$ to be arbitrarily large. We give a range of applications and examples, such as any group $G$ with $\beta_1(G)\geq 2$ that is $F_n$-by-$\mathbf Z$ for $F_n$ the non-abelian free group of rank $n$ is also $F_m$-by-$\mathbf Z$ 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.

#### Article information

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

Dates
First available in Project Euclid: 1 October 2007

https://projecteuclid.org/euclid.jmsj/1191247591

Digital Object Identifier
doi:10.2969/jmsj/05920351

Mathematical Reviews number (MathSciNet)
MR2325689

Zentralblatt MATH identifier
1124.57001

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1191247591

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