  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.

Geometric Topology;Group Theory