Title: Peripheral subgroups of Kleinian groups Show Chinese title

Title: 克莱因群的外围子群

Abstract: The conformal boundary of a hyperbolic $3$-manifold $M$ is a union of Riemann surfaces. If any of these Riemann surfaces has a nontrivial Teichm\"uller space, then the hyperbolic metric of $M$ can be deformed quasi-isometrically. These deformations correspond to a small pertubations in the matrices of the holonomy group of $M$, which together give an island of discrete representations around the identity map in $ X=\mathrm{Hom}(\pi_1(M), \mathsf{PSL}(2,\mathbb{C})) $. Determining the extent of this island is a hard problem. If $M$ is geometrically finite and its convex core boundary is pleated only along simple closed curves, then we cut up its conformal boundary in a way governed by the pleating combinatorics to produce a fundamental domain for $ \pi_1(M) $ that is combinatorially stable under small deformations, even those which change the pleated structure. We give a computable region in $X$, cut out by polynomial inequalities over $\mathbb{R}$, within which this fundamental domain is valid: all the groups in the region have peripheral structures that look `coarsely similar', in that they come from real-algebraically deforming a fixed conformal polygon and its side-pairings. The union of all these regions for different pleating laminations gives a countable cover, with sets of controlled topology, of the entire quasi-isometric deformation space of $ \pi_1(M) $ -- which is known to be topologically wild. Show Chinese abstract

Abstract: 双曲$3$流形$M$的共形边界是黎曼曲面的并集。 如果这些黎曼曲面中的任何一个具有非平凡的Teichmüller空间，那么双曲度量$M$可以进行准等距变形。 这些变形对应于$M$的Holonomy群矩阵的小扰动，这些扰动一起在$ X=\mathrm{Hom}(\pi_1(M), \mathsf{PSL}(2,\mathbb{C})) $中给出一个围绕恒等映射的离散表示岛屿。 确定这个岛屿的范围是一个困难的问题。 如果$M$是几何有限的，并且其凸核心边界仅沿着简单闭曲线褶皱，那么我们根据褶皱组合学的方式切割其共形边界，以生成一个对于$ \pi_1(M) $的基本区域，在小变形下，即使改变褶皱结构，该基本区域在组合上也是稳定的。 我们在$X$中给出一个可计算的区域，该区域由$\mathbb{R}$上的多项式不等式切割而成，在该区域内这个基本域是有效的：该区域内的所有群都有周边结构，它们在某种粗略意义上是相似的，即它们来自于固定共形多边形及其边配对的实代数变形。 不同折叠层的这些区域的并集给出了$ \pi_1(M) $的整个拟等距变形空间的一个可数覆盖，其集合具有受控的拓扑结构——该空间已知在拓扑上是复杂的。