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Conformal Geometry and Dynamics

ISSN 1088-4173



Nielsen equivalence in mapping tori over the torus

Author: Ian Biringer
Journal: Conform. Geom. Dyn. 21 (2017), 105-110
MSC (2010): Primary 57M07
Published electronically: March 13, 2017
MathSciNet review: 3622115
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Abstract: We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb {Z}$ and if $G_A=\mathbb {Z}^2 \rtimes _A \mathbb {Z}$ is generated by two elements, then there is a single Nielsen equivalence class of $2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm \left (\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in which case there are two.

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Additional Information

Ian Biringer
Affiliation: Boston College, Department of Mathematics, 140 Commonwealth Ave, Chestnut Hill, MA 02467

Keywords: Nielsen equivalence, Farey graph
Received by editor(s): October 27, 2016
Received by editor(s) in revised form: February 23, 2017
Published electronically: March 13, 2017
Additional Notes: The author was partially supported by NSF grant DMS 1611851
Article copyright: © Copyright 2017 American Mathematical Society