Shapes of hyperbolic triangles and once-punctured torus groups

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 X.~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.