Shapes of hyperbolic triangles and once-punctured torus groups
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 $φ\in(0,π)\setminus\mathbb{Q}π$, 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 $\mathfrak{C}_θ$ 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 $ξ\in\mathfrak{C}_θ$ 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.