AbstractLet $$\Delta $$
Δ
be a hyperbolic triangle with a fixed area $$\varphi $$
φ
. We prove that for all but countably many $$\varphi $$
φ
, generic choices of $$\Delta $$
Δ
have the property that the group generated by the $$\pi $$
π
-rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$
φ
∈
(
0
,
π
)
\
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 $$\mathfrak {C}_\theta $$
C
θ
of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$
θ
=
2
(
π
-
φ
)
, and answer an analogous question for the holonomy map $$\rho _\xi $$
ρ
ξ
of such a hyperbolic structure $$\xi $$
ξ
. In an appendix by Gao, concrete examples of $$\theta $$
θ
and $$\xi \in \mathfrak {C}_\theta $$
ξ
∈
C
θ
are given where the image of each $$\rho _\xi $$
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.
Shapes of hyperbolic triangles and once-punctured torus groups
