Please use this identifier to cite or link to this item: https://doi.org/10.1007/s00209-021-02745-3
Title: Shapes of hyperbolic triangles and once-punctured torus groups
Authors: Kim, Sang-hyun
Koberda, Thomas
Lee, Jaejeong
Ohshika, Ken'ichi
Tan, Ser Peow 
Gao, Xinghua
Keywords: Fuchsian groups
Kleinian groups
Issue Date: 3-May-2021
Publisher: Springer Science and Business Media Deutschland GmbH
Citation: Kim, Sang-hyun, Koberda, Thomas, Lee, Jaejeong, Ohshika, Ken'ichi, Tan, Ser Peow, Gao, Xinghua (2021-05-03). Shapes of hyperbolic triangles and once-punctured torus groups. Mathematische Zeitschrift 299 (3-Apr) : 2103-2130. ScholarBank@NUS Repository. https://doi.org/10.1007/s00209-021-02745-3
Rights: Attribution 4.0 International
Abstract: Let Δ be a hyperbolic triangle with a fixed area φ. We prove that for all but countably many φ, generic choices of Δ have the property that the group generated by the π-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all φ∈ (0 , π) \ Qπ, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space Cθ of singular hyperbolic metrics on a torus with a single cone point of angle θ= 2 (π- φ) , and answer an analogous question for the holonomy map ρξ of such a hyperbolic structure ξ. In an appendix by Gao, concrete examples of θ and ξ∈ Cθ are given where the image of each ρξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. © 2021, The Author(s).
Source Title: Mathematische Zeitschrift
URI: https://scholarbank.nus.edu.sg/handle/10635/232198
ISSN: 0025-5874
DOI: 10.1007/s00209-021-02745-3
Rights: Attribution 4.0 International
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