实双曲5空间中的迹坐标与局部Fenchel--Nielsen参数
Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space
- Indian Institute of Science Education and Research (IISER) Mohali（印度科学教育与研究学院莫哈利分校）
- Dr. Sagar Kalane Academy（萨加尔·卡兰博士学院）
- Kalna College（卡尔纳学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该文将Fenchel--Nielsen理论推广到实双曲5空间，通过四元数表示给出闭曲面群表示的局部坐标，得到30g-30个实参数。
AI中文摘要：
我们研究闭可定向曲面基本群到 \\(\mathrm{SL}(2,\mathbb{H})\\) 中的表示，其射影化 \\(\mathrm{PSL}(2,\mathbb{H})\\) 是实双曲 \\(5\\) 空间保定向等距群，目的是发展 Fenchel--Nielsen 理论的四元数类比。我们给出了具有不相交不动点集的正则 loxodromic 元素对的规范形式，并证明在开稠密的一般轨迹上，相应的模空间是 \\(15\\) 维的。我们还关联了十五个字的迹，其微分在开稠密子集上线性无关，因此在该处给出局部实解析坐标。对于具有指定正则 loxodromic 边界共轭类的裤子，我们证明了相对形变空间局部是 \\(6\\) 维的。将此内部裤子数据与三维边界共轭数据以及三维中心化子粘合自由度相结合，给出了闭曲面群表示的局部参数分解。对于亏格为 \\(g\\) 的闭曲面，这产生了 \\(30g-30\\) 个实参数。
英文摘要：
We study representations of fundamental groups of closed orientable surfaces into \(\mathrm{SL}(2,\mathbb{H})\), whose projectivization \(\mathrm{PSL}(2,\mathbb{H})\) is the group of orientation-preserving isometries of real hyperbolic \(5\)-space, with the aim of developing a quaternionic analogue of Fenchel--Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is \(15\)-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there. For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally \(6\)-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus \(g\), this yields \(30g-30\) real parameters.
