Profinite topology on non-positively curved groups

几何群论是一个广阔的数学领域，它结合了代数、分析、几何和拓扑学的思想，对所有这些学科都做出了重要贡献。这一领域在过去的20年中得到了迅速的发展，并受到全世界数学家的欢迎。几何群论的一个主要主题是对非正弯曲群的研究。如果一个群G通过变换（以足够好的方式）作用于空间X，它是非正弯曲的，其几何形状类似于欧几里德空间或双曲空间的几何形状。在这种作用存在的情况下，X的性质给出了很多关于G结构的信息，反之亦然。研究无限离散群G最自然的方法之一就是看它的有限商。然而，一般来说，很多关于G的信息不能通过这种方式恢复；例如，存在无限群，它们根本没有非平凡有限商。这是引入G的以下两个性质的经典原因。G群被称为剩余有限的，如果对于G中的任意两个不同的元素x，y，G的有限商群Q使得x和y的像在Q中是不同的，如果对于G中的任意两个非共轭元素x，y存在从G到有限群Q的同态，并将x和y映射为Q的非共轭元素，则G被称为共轭可分性。残差有限性和共轭可分性分别是词和共轭问题的可解性的自然组合类似物。实际上，Mal'cev的一个经典定理断言，一个有限呈现的剩余有限群具有可解的词问题。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是一个困难得多的任务。直到最近，婚配可分离性只在少数家族或群体中为人所知。本课题旨在证明大类别非正弯曲群的共轭可分性，并建立它们的自同构群的剩余有限性。其结果将提高我们对非正弯曲群的几何和代数性质之间联系的理解，并将对几何群论中突出的开放问题有所启示。在最近的一篇论文中，PI证明了直角Artin群是非正弯曲群的一个重要子类，并且它们的所有有限指数子群都是共轭可分的。将该代数定理与Haglund和Wise的几何结果结合起来，为共轭可分群提供了大量的新例子，其意义更加明显。在这项工作中，PI发现并开发了一些强大的研究群剩余性质的工具。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，以建立更多群的共轭可分性。第二部分研究了某些非正弯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的共轭可分性与Out(G)的剩余有限性之间的联系，以及由Bowditch， Levitt和PI-Osin得到的关于相对双曲群的自同构的结构结果。

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.