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，并对所有这些学科做出了重要贡献。这一领域在过去的20年里得到了迅速发展，并在世界各地的数学家中受到欢迎。几何群论的主要主题之一是研究非正曲群。群G是非正曲的，如果它通过变换（以足够好的方式）作用在空间X上，其几何类似于欧几里得或双曲空间的几何。在存在这样的作用时，X的性质给出了关于G的结构的许多信息，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子群。然而，一般来说，关于G的大部分信息不能以这种方式恢复;例如，存在根本没有非平凡有限继元的无限群。这是引入G的以下两个性质的经典理由。称群G是剩余有限的，如果对于G中的任意两个不同的元素x，y，存在G的有限对偶群Q，使得x和y的像在Q中不同。称G为共轭可分的，如果对G中的任意两个非共轭元x，y，存在G到有限群Q的同态，该同态将x，y映射到Q的非共轭元上.剩余有限性和共轭可分性分别是字问题和共轭问题可解性的自然组合类比。事实上，一个经典定理的马尔切夫断言，一个randompresented剩余有限[共轭可分]组有可解的字问题[共轭问题]。许多群很容易被证明是剩余有限的;另一方面，证明群是共轭可分的是一项困难得多的任务。直到最近，共轭可分性只在少数几个群族中才被发现。这个项目的目标是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。它的结果将提高我们的理解之间的连接几何和代数性质的非积极弯曲的群体，并将揭示一些悬而未决的问题在几何群论。在最近的一篇论文中，PI证明，直角阿廷群，形成一个重要的子类的非积极弯曲的群体，和所有他们的有限指数子群是共轭可分的。这个代数定理的意义变得清晰后，结合它与几何结果Haglund和怀斯，它提供了大量的新的例子共轭可分群。PI在这项工作中发现并开发了几个研究群的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，以建立更多的群的共轭可分性。第二部分研究了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，给出群G的共轭可分性与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.