对称空间 S L ( n , R ) / S O ( n , R ) 中塞尔伯格平分线的几何学 $SL(n,\mathbb {R})/SO(n,\mathbb {R})$

IF 1 2区 数学 Q1 MATHEMATICS
Yukun Du
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引用次数: 0

摘要

我研究了与李群 S L ( n , R ) $SL(n,\mathbb{R})$相关的对称空间的几个问题。这些问题与基于波恩卡莱基本多面体定理的算法有关,该算法旨在确定 S L ( n , R ) $SL(n,\mathbb {R})$ 子群的广义几何有限性属性。该算法类似于双曲空间中的原始算法,而黎曼距离则由 S L ( n , R ) $SL(n,\mathbb {R})$ 不变量前对称取代。本文的主要结果有两部分。在第一部分中,我重点讨论了将波恩卡莱算法推广到我的对称空间时出现的问题。我描述并实现了一种算法,它可以计算有限边多面体的面集结构,并构造超平面之间的类角函数。在第二部分中,我将进一步研究与我的对称空间中的超平面和 Dirichlet-Selberg 域相关的问题。我建立了超平面不相交的几个标准,并根据它们的 Dirichlet-Selberg 域是否是有限边,对 S L ( 3 , R ) $SL(3,\mathbb {R})$ 的特定阿贝尔子群进行了分类。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Geometry of Selberg's bisectors in the symmetric space S L ( n , R ) / S O ( n , R ) $SL(n,\mathbb {R})/SO(n,\mathbb {R})$

I study several problems about the symmetric space associated with the Lie group S L ( n , R ) $SL(n,\mathbb {R})$ . These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of S L ( n , R ) $SL(n,\mathbb {R})$ . The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an S L ( n , R ) $SL(n,\mathbb {R})$ -invariant premetric. The main results of this paper are twofold. In the first part, I focus on questions that occurred in generalizing Poincaré's Algorithm to my symmetric space. I describe and implement an algorithm that computes the face-poset structure of finitely sided polyhedra, and construct an angle-like function between hyperplanes. In the second part, I study further questions related to hyperplanes and Dirichlet–Selberg domains in my symmetric space. I establish several criteria for the disjointness of hyperplanes and classify particular Abelian subgroups of S L ( 3 , R ) $SL(3,\mathbb {R})$ based on whether their Dirichlet–Selberg domains are finitely sided or not.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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