权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Profinite topology on non-positively curved groups

非正曲群上的有限拓扑

基本信息

批准号：
EP/H032428/1
负责人：
Ashot Minasyan
金额：
$ 12.86万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH032428%2F1
关键词：
Profinite topology non positively curved

项目摘要

Geometric Group Theory is a vast area of Mathematics that combines ideas from Algebra, Analysis, Geometry and Topology and makes important contributions to all of these subjects. This area has been rapidly developing during the last 20 years, and has become popular among mathematicians all around the world. One of the principal themes of Geometric Group Theory is the study of non-positively curved groups. A group G is non-positively curved if it acts by transformations (in a sufficiently good manner) on a space X, whose geometry is similar to the geometry of a Euclidean or Hyperbolic space. In the presence of such an action, the properties of X give a lot of information about the structure of G and vice-versa.One of the most natural ways to study an infinite discrete group G is to look at its finite quotients. However, in general much of the information about G cannot be recovered this way; e.g., there exist infinite groups which have no non-trivial finite quotients at all. This is the classical reason for introducing the following two properties of G. The group G is said to be residually finite if for any two distinct elements x,y in G, there is a finite quotient-group Q of G such that the images of x and y are distinct in Q. And G is called conjugacy separable if for any two non-conjugate elements x,y in G there is a homomorphisms from G to finite group Q which maps x and y to non-conjugate elements of Q. Residual finiteness and conjugacy separability are natural combinatorial analogues of solvability of the word and conjugacy problems respectively. Indeed, a classical theorem of Mal'cev asserts that a finitely presented residually finite [conjugacy separable] group has solvable word problem [conjugacy problem]. Many groups are easily shown to be residually finite; on the other hand, proving that a group is conjugacy separable is a much more difficult task. Until recently, conjugacy separability was known for only a few families of groups.The proposed project aims to prove conjugacy separability for large classes of non-positively curved groups and establish residual finiteness for their automorphism groups. Its outcome will improve our understanding of the connection between geometric and algebraic properties of non-positively curved groups, and will shed some light on outstanding open problems in Geometric Group Theory.In a recent paper the PI proved that right angled Artin groups, forming an important subclass of non-positively curved groups, and all of their finite index subgroups are conjugacy separable. The significance of this algebraic theorem becomes clear after combining it with geometric results of Haglund and Wise, which provides an abundance of new examples of conjugacy separable groups. Several powerful tools for studying residual properties of a group were discovered and developed by the PI in this work. In the first part of the project we intend to use these tools and introduce new ones in order to establish conjugacy separability of many more groups. The second part will be dedicated to investigation of residual finiteness of outer automorphism groups for certain non-positively curved groups. Our approach here will be based on the theorem of Grossman, providing a connection between conjugacy separability of a group G and residual finiteness of Out(G), together with the structure results about automorphisms of relatively hyperbolic groups which were obtained by Bowditch, Levitt and the PI-Osin.

几何群论是一个广泛的数学领域，它结合了代数、分析、几何和拓扑学的思想，并对所有这些学科做出了重要贡献。在过去的20年里，这个领域得到了迅速的发展，并在世界各地的数学家中变得流行起来。几何群论的主要主题之一是研究非正曲群。如果群G通过变换(以足够好的方式)作用于空间X，且X的几何类似于欧几里得或双曲空间的几何，则群G是非正弯曲的。在这样的作用下，X的性质给出了许多关于G的结构的信息，反之亦然。研究无限离散群G的最自然的方法之一是研究它的有限商。然而，一般而言，关于G的许多信息不能以这种方式恢复；例如，存在根本没有非平凡有限商的无限群。这是引入G的两个性质的经典原因。群G称为剩余有限的，如果对于G中的任何两个不同的元素x，y，存在G的有限商群Q使得x和y的像在Q中是不同的。如果G中的任何两个非共轭元素x，y存在从G到有限群Q的同态，它将x和y映射到Q的非共轭元素，则称G是共轭可分的。剩余有限性和共轭可分性分别是字的可解性和共轭问题的自然的组合模拟。事实上，Mal‘cev的一个经典定理断言，有限表示的剩余有限[共轭可分]群存在可解字问题[共轭问题]。许多群很容易被证明是剩余有限的；另一方面，证明一个群是共轭可分的要困难得多。直到最近，共轭可分性仅为少数几个群族所知，该项目旨在证明大类非正曲群的共轭可分性，并建立它们的自同构群的剩余有限性。它的结果将加深我们对非正曲群几何性质和代数性质之间关系的理解，并将有助于揭示几何群论中的一些未决问题。在最近的一篇文章中，PI证明了构成非正曲群的一个重要子类的直角Artin群及其所有有限指标子群是共轭可分的。这一代数定理与Haglund和Wise的几何结果相结合，给出了大量共轭可分群的新例子，其意义变得更加明显。在这项工作中，PI发现并开发了几个研究群剩余性质的强大工具。在项目的第一部分，我们打算使用这些工具并引入新的工具，以建立更多组的共轭可分性。第二部分将致力于研究某些非正曲群的外自同构群的剩余有限性。我们的方法将基于Grossman定理，提供群G的共轭可分性和Out(G)的剩余有限性之间的联系，以及Bowditch，Levitt和Pi-Osin关于相对双曲群的自同构的结构结果。