PU(n,1) 中的对合与换位子长度
Involution and Commutator Length in PU(n,1)
- School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明复双曲等距群 PU(n,1)（n≥2）的对合长度为 4，且每个元素均为单换位子，验证了 Djoković 猜想 A。
AI中文摘要：
我们研究复双曲空间的全纯等距分解为全纯对合的问题。我们证明，对于每个 n>=3，PU(n,1) 的对合长度为 4。这改进了 Paupert--Will 的高维上界 8，以及由 Bünger 线性分解定理隐含的射影上界 5，达到最优值 4。结合 Paupert--Will 的二维结果，这表明对于每个 n>=2，PU(n,1) 的对合长度为 4。证明的核心是平方根的三对合分解定理：PU(n,1) 中的每个 g 都有一个对合长度至多为 3 的平方根，并且这个一致界是最优的。关键构造首先在二维或三维不定块上进行，然后通过正定正交补上的必要且充分的谱配对准则在高维中完成。四因子下界由点处的复反射提供。作为推论，PU(n,1) 的每个元素都是单个换位子，并且换位子表示中的两个元素可以被同一个非平凡全纯对合同时逆转。这将 Paupert--Will 的二维单换位子结果推广到每个 n>=2，从而验证了 Djoković 猜想 A 对族 PU(n,1)（n>=2）成立。
英文摘要：
We study decompositions of holomorphic isometries of complex hyperbolic space into holomorphic involutions. We prove that, for every n>=3, the involution length of PU(n,1) is 4. This improves the higher-dimensional upper bound 8 of Paupert--Will, as well as the projective upper bound 5 implied by Bünger's linear decomposition theorem, to the optimal value 4. Combined with the two-dimensional result of Paupert--Will, this shows that the involution length of PU(n,1) is 4 for every n>=2. The core of the proof is a three-involution decomposition theorem for square roots: every g in PU(n,1) has a square root of involution length at most 3, and this uniform bound is optimal. The key construction is carried out first on a two- or three-dimensional indefinite block and then completed in higher dimensions by a necessary and sufficient spectral pairing criterion on the positive-definite orthogonal complement. The four-factor lower bound is provided by complex reflections in a point. As a consequence, every element of PU(n,1) is a single commutator, and the two elements in the commutator representation can be simultaneously reversed by the same nontrivial holomorphic involution. This extends the two-dimensional single-commutator result of Paupert--Will to every n>=2 and thereby verifies Djoković's Conjecture A for the family PU(n,1), n>=2.
