Shapes of hyperbolic triangles and once-punctured torus groups

Abstract

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.