<title>Abstract</title>Let <inline-formula><alternatives><tex-math>$$\Delta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Δ</mml:mi>
</mml:math></alternatives></inline-formula> be a hyperbolic triangle with a fixed area <inline-formula><alternatives><tex-math>$$\varphi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>φ</mml:mi>
</mml:math></alternatives></inline-formula>. We prove that for all but countably many <inline-formula><alternatives><tex-math>$$\varphi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>φ</mml:mi>
</mml:math></alternatives></inline-formula>, generic choices of <inline-formula><alternatives><tex-math>$$\Delta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Δ</mml:mi>
</mml:math></alternatives></inline-formula> have the property that the group generated by the <inline-formula><alternatives><tex-math>$$\pi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>π</mml:mi>
</mml:math></alternatives></inline-formula>-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all <inline-formula><alternatives><tex-math>$$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$</tex-math><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></alternatives></inline-formula>, 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 <inline-formula><alternatives><tex-math>$$\mathfrak {C}_\theta $$</tex-math><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></alternatives></inline-formula> of singular hyperbolic metrics on a torus with a single cone point of angle <inline-formula><alternatives><tex-math>$$\theta =2(\pi -\varphi )$$</tex-math><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></alternatives></inline-formula>, and answer an analogous question for the holonomy map <inline-formula><alternatives><tex-math>$$\rho _\xi $$</tex-math><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></alternatives></inline-formula> of such a hyperbolic structure <inline-formula><alternatives><tex-math>$$\xi $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>ξ</mml:mi>
</mml:math></alternatives></inline-formula>. In an appendix by Gao, concrete examples of <inline-formula><alternatives><tex-math>$$\theta $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>θ</mml:mi>
</mml:math></alternatives></inline-formula> and <inline-formula><alternatives><tex-math>$$\xi \in \mathfrak {C}_\theta $$</tex-math><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></alternatives></inline-formula> are given where the image of each <inline-formula><alternatives><tex-math>$$\rho _\xi $$</tex-math><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></alternatives></inline-formula> is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.