Suppose that <i></i>₀ is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup <i>G</i> of the Mbius group of <i>S³</i> = the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy <i></i>_<i>t</i>, 0 <i>t</i> 1, from <i></i>₀ to a round circle <i></i>₁.

Given a compact orientable surface with finitely many punctures \Sigma , let $\Cal S (\Sigma) $ be the set of isotopy classes of essential unoriented simple closed curves in \Sigma . We determine a complete set of relations for a function from $\Cal S (\Sigma) $ to $\bold R $ to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on \Sigma . As a conse quence, the Teichmller space of hyperbolic metrics with geodesic boundary and cusp ends on \Sigma is reconstructed from an intrinsic $(\bold QP^ 1, PSL (2,\bold Z)) $ structure on $\Cal S (\Sigma) $.

Given a loop on a surface, its homotopy class can be specified as a word consisting of letters representing the homotopy group generators. One of the interesting problems is how to compute the shortest word for a given loop. This is an NP-hard problem in general. However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently. In this paper, we propose an efficient algorithm to compute the shortest words for loops given on triangulated surface meshes. The design of this algorithm is inspired and guided by the work of Dehn and BirmanSeries. In support of the shortest word algorithm, we also propose efficient algorithms to compute shortest paths and shortest loops under hyperbolic metrics using a novel