RUI: Twisted Conjugacy, Reidemeister Number and Thompson's Groups

汤普森群F出现在数学的各个分支中，从逻辑学到代数到同伦论和群论。它可以理解为具有标准有限生成集的有限或无限生成群。作者提出通过研究群相对于任意发电集的度量性质来消除对该发电集的依赖。在之前工作的基础上，她将研究这个群是否是自动的，以及F的标准推广F(p)的准等距分类。在此授权下提出的剩余工作涉及扭曲共轭类和群自同构的Reidemeister数。如果f是紧流形的自映射，则基群上的诱导映射的扭曲共轭类的个数称为Reidemeister数R(f)。Reidemeister数为f提供了Nielsen数N(f)的上界，这个上界很难计算。如果流形的任何自映射的Reidemeister数是无限的，这就消除了一个特殊类型的定理等于N(f)和R(f)的可能性。首席研究员建议扩展具有这一性质的（基本）群的类别，并建立在先前的工作基础上，看看它在准等距下何时是不变的。她还将研究扭曲共轭类的几何解释，以及这一性质与子群可分性的关系。主要研究者在几何群论领域提出了几个项目。这个领域从几何的角度研究被称为群的基本数学对象。对称群是数学群的一个典型例子。例如，对对称群的研究被用来辨别DNA的形状是双螺旋结构。首席研究员密切研究一个特定的群体，以首次定义该群体的研究人员命名为汤普森群体。汤普森的群可以用代数、几何和分析的方法来理解，允许人们从不同的角度来研究它。几何视角将每组元素等同于一对“树状”图，这种方法将这组元素与理论计算机科学中的问题联系起来。通过这种方式，它提供了抽象数学的有趣的跨学科应用，直接应用于转换数据以最大化某些计算机搜索算法的效率的问题。首席研究员提出了几个与理解这个群体的几何形状有关的问题。她提案的其余部分涉及几何群论和拓扑学交叉的几个问题。

Thompson's group F appears in varied branches of mathematics, from logic to algebra to homotopy theory and group theory. It can be understood either as a finitely or infinitely generated group, with a standard finite generating set. The principal investigator proposes to remove the dependence on this generating set by studying metric properties of the group with respect to arbitrary generating sets. Building on previous work, she will investigate whether this group is automatic, as well as a quasi-isometry classification for the standard generalizations F(p) of F. The remaining work proposed under this grant concerns twisted conjugacy classes and the Reidemeister number of a group automorphism. If f is a selfmap of a compact manifold, then the number of twisted conjugacy classes of the induced map on the fundamental group is called the Reidemeister number R(f). The Reidemeister number provides an upper bound on the Nielsen number N(f) for f, which is difficult to compute. If the Reidemeister number is infinite for any selfmap of the manifold, this eliminates the possibility of a special type of theorem equating N(f) and R(f). The principal investigator proposes to extend the classes of (fundamental) groups with this property, and build on prior work to see when it is invariant under quasi-isometry. She will also investigate a geometric interpretation of twisted conjugacy classes, and the relationship of this property to subgroup separability.The principal investigator proposes several projects in the areaof geometric group theory. This field studies fundamental mathematical objects called groups from a geometric point of view. Symmetry groups are a typical example of mathematical groups. The study of symmetry groups was used, for example, to discern that the shape of DNA was a double helix. The principal investigator studies closely one particular group, named Thompson's group after the researcher who first defined it. Thompson's group can be understood algebraically, geometrically and analytically, allowing one to study it from differing viewpoints. The geometric perspective equates each group element with a pair of "tree-like" graphs, an approach which relates this group to questions in theoretical computer science. In this way it provides an interesting interdisciplinary application of abstract mathematics, with direct applications to the problem of transforming data to maximize the efficiency of some computer search algorithms. The principal investigator proposes several problems related to understanding the geometry of this group. The remainder of her proposal concerns several problems at the intersection of geometric group theory and topology.