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Conformal Geometry and Dynamics

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Nielsen equivalence in mapping tori over the torus


Author: Ian Biringer
Journal: Conform. Geom. Dyn. 21 (2017), 105-110
MSC (2010): Primary 57M07
DOI: https://doi.org/10.1090/ecgd/308
Published electronically: March 13, 2017
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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
Email: ianbiringer@gmail.com

DOI: https://doi.org/10.1090/ecgd/308
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