Let \(\Delta \) be a hyperbolic triangle with a fixed area \(\varphi \). We prove that for all but countably many \(\varphi \), generic choices of \(\Delta \) have the property that the group generated by the \(\pi \)-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all \(\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi \), 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 \(\mathfrak {C}_\theta \) of singular hyperbolic metrics on a torus with a single cone point of angle \(\theta =2(\pi -\varphi )\), and answer an analogous question for the holonomy map \(\rho _\xi \) of such a hyperbolic structure \(\xi \). In an appendix by Gao, concrete examples of \(\theta \) and \(\xi \in \mathfrak {C}_\theta \) are given where the image of each \(\rho _\xi \) is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

令\（\ Delta \）是一个固定面积\（\ varphi \）的双曲三角形。我们证明，除了几乎所有的所有\（\ varphi \）之外，\（\ Delta \）的泛型选择都具有以下性质：由\（\ pi \）-旋转围绕三角形边的中点生成的组不承认任何非平凡的关系。相比之下，对于所有\（\ varphi \ in（0，\ pi）{\ setminus} \ mathbb {Q} \ pi \），我们显示了一组紧密的三角形确实具有非平凡的关系，这在一般情况下映射为双曲线翻译。为了建立这一事实，我们研究了具有单个圆锥角的圆环上奇异双曲度量的变形空间\（\ mathfrak {C} _ \ theta \）\（\ theta = 2（\ pi-\ varphi）\），并回答类似双曲结构\（\ xi \）的完整图\（\ rho _ \ xi \）的类似问题。在高的附录中，给出了\（\ theta \）和\（\ xi \ in \ mathfrak {C} _ \ theta \）中的具体示例，其中每个\（\ rho _ \ xi \）的图像都是有限的呈现的，非自由的和无扭转的；实际上，这些图像对于封闭的双曲3型流形的基本群将是同构的。