Shapes of hyperbolic triangles and once-punctured torus groups

2020/06/04 by Sang-Hyun Kim, Kim, Sang-hyun, Thomas Koberda +8
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2006.02621

openalex publication_date 2020/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfrakC_θ 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 ρ_ξ of such a hyperbolic structure ξ. In an appendix by X.~Gao, concrete examples of θ and ξ∈\mathfrakC_θ are given where the image of each ρ_ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.

Citations

• The invariant theory of n × n matrices

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