{"title":"Geometry of Selberg's bisectors in the symmetric space \n \n \n S\n L\n (\n n\n ,\n R\n )\n /\n S\n O\n (\n n\n ,\n R\n )\n \n $SL(n,\\mathbb {R})/SO(n,\\mathbb {R})$","authors":"Yukun Du","doi":"10.1112/jlms.12992","DOIUrl":null,"url":null,"abstract":"<p>I study several problems about the symmetric space associated with the Lie group <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>L</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>,</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$SL(n,\\mathbb {R})$</annotation>\n </semantics></math>. These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>L</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>,</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$SL(n,\\mathbb {R})$</annotation>\n </semantics></math>. The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>L</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>,</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$SL(n,\\mathbb {R})$</annotation>\n </semantics></math>-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 <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>L</mi>\n <mo>(</mo>\n <mn>3</mn>\n <mo>,</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$SL(3,\\mathbb {R})$</annotation>\n </semantics></math> based on whether their Dirichlet–Selberg domains are finitely sided or not.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12992","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
I study several problems about the symmetric space associated with the Lie group . These problems are connected to an algorithm based on Poincaré's Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of . The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an -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 based on whether their Dirichlet–Selberg domains are finitely sided or not.
对称空间 S L ( n , R ) / S O ( n , R ) 中塞尔伯格平分线的几何学 $SL(n,\mathbb {R})/SO(n,\mathbb {R})$
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。
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.