1 research outputs found
Shapes of hyperbolic triangles and once-punctured torus groups
Let be a hyperbolic triangle with a fixed area . We prove
that for all but countably many , generic choices of have the
property that the group generated by the --rotations about the midpoints
of the sides of the triangle admits no nontrivial relations. By contrast, we
show for all , 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
of singular hyperbolic metrics on a torus with a single
cone point of angle , and answer an analogous question
for the holonomy map of such a hyperbolic structure . In an
appendix by X.~Gao, concrete examples of and
are given where the image of each is
finitely presented, non-free and torsion-free; in fact, those images will be
isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.Comment: 32 pages. To appear in Math.