Indiscrete representations of discrete groups

我们完全确定了2-桥链的2-桥球上的那些简单环在环补中是零同伦的或周边的。我们还完全确定了2-桥环的2-桥球面上的两个简单环何时在环补中同伦。作为这些结果的一个应用，我们建立了两桥环的一个变形的McShane‘S恒等式，它给出了一个用闭测地线的复平移长度来表示两桥环的尖点模数的公式

We completely determined those simple loops on the 2-bridge spheres of 2-bridge links to be null-homotopic or peripheral in the link complements. We also completely determined when two simple loops on the 2-bridge spheres of 2-bridge links to be homotopic in the link complements. As an application of these results, we established a variation of McShane' s identity for 2-bridge links, which gives a formula to express the modulus of the cusp of a 2-bridge link in terms of the complex translation lengths of closed geodesics