Heegaard structures and geometric structures of 3-manifolds

我们集中研究了一次穿刺环面，这是最简单的双曲曲面，相信它会让我们对一般双曲曲面有更深的理解，得到了以下结果。(1)对Jorgensen关于拟狐猴刺穿环面群的理论进行了完整的描述和证明。(2)我们发现了以下两种与环面束在圆上穿孔相关的镶嵌之间的密切关系；Cannon-Thurston-Dicks分形镶嵌和由典型分解引起的尖三角剖分。我们还提出了一个关于穿孔表面束在圆上正则分解的猜想。(3)给出了2桥结的桥球上在结补上为零同伦的基本简单环的完整刻划。

We have concentrated on the study of the once-punctured torus, the simplest hyperbolic surface, believing that it would bring us to deep understanding of general hyperbolic surfaces, and obtained the following results. (1) We gave a complete description and proof to Jorgensen's theory on quasifuchsian punctured torus groups. (2) We found an intimate relation between the following two tessellations associated with a punctured torus bundles over the circle ; the Cannon-Thurston-Dicks fractal tessellation and the cusp triangulation induced by the canionical decomposition. We also proposed a conjecture concerning the canonical decompositions of punctured surface bundles over the circle. (3) We gave a complete characterization of those essential simple loops on the bridge sphere of a 2-bridge knot which are null-homotopic in the knot complement.