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Hyperbolic one-relator groups

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  06 May 2024

Marco Linton Open the ORCID record for Marco Linton [Opens in a new window]
Marco Linton*
Affiliation:
Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford, OX2 6GG, United Kingdom
*
e-mail: marco.linton@maths.ox.ac.uk
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Abstract

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

Keywords

One-relator groupshyperbolic groupsfree groupsGersten’s conjectureMagnus hierarchy

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups 20F05: Generators, relations, and presentations
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 27
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

While carrying out part of this work, the author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 850930).

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