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.
Publié le : 2007-04-14
Classification:
mapping torus,
BNS invariant,
Alexander polynomial,
57M05,
20F65,
57N10
@article{1191247591,
author = {BUTTON, Jack O.},
title = {Mapping tori with first Betti number at least two},
journal = {J. Math. Soc. Japan},
volume = {59},
number = {1},
year = {2007},
pages = { 351-370},
language = {en},
url = {http://dml.mathdoc.fr/item/1191247591}
}
BUTTON, Jack O. Mapping tori with first Betti number at least two. J. Math. Soc. Japan, Tome 59 (2007) no. 1, pp. 351-370. http://gdmltest.u-ga.fr/item/1191247591/