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• W3159941549 abstract "Abstract Let $$Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>Δ</mml:mi></mml:math> be a hyperbolic triangle with a fixed area $$varphi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>φ</mml:mi></mml:math> . We prove that for all but countably many $$varphi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>φ</mml:mi></mml:math> , generic choices of $$Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>Δ</mml:mi></mml:math> have the property that the group generated by the $$pi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>π</mml:mi></mml:math> -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$varphi in (0,pi ){setminus }mathbb {Q}pi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow><mml:mi>φ</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo>)</mml:mo><mml:mo></mml:mo><mml:mi>Q</mml:mi><mml:mi>π</mml:mi></mml:mrow></mml:math> , 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 $$mathfrak {C}_theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>C</mml:mi><mml:mi>θ</mml:mi></mml:msub></mml:math> of singular hyperbolic metrics on a torus with a single cone point of angle $$theta =2(pi -varphi )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>(</mml:mo><mml:mi>π</mml:mi><mml:mo>-</mml:mo><mml:mi>φ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> , and answer an analogous question for the holonomy map $$rho _xi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>ρ</mml:mi><mml:mi>ξ</mml:mi></mml:msub></mml:math> of such a hyperbolic structure $$xi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>ξ</mml:mi></mml:math> . In an appendix by Gao, concrete examples of $$theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>θ</mml:mi></mml:math> and $$xi in mathfrak {C}_theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow><mml:mi>ξ</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>θ</mml:mi></mml:msub></mml:mrow></mml:math> are given where the image of each $$rho _xi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msub><mml:mi>ρ</mml:mi><mml:mi>ξ</mml:mi></mml:msub></mml:math> is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds." @default.
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• W3159941549 date "2021-05-03" @default.
• W3159941549 modified "2023-09-27" @default.
• W3159941549 title "Shapes of hyperbolic triangles and once-punctured torus groups" @default.
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• W3159941549 doi "https://doi.org/10.1007/s00209-021-02745-3" @default.
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