{"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":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12992","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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:
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.