Geometry of Subgroups

群是一个代数对象，它是一个几何对象的对称的集合。例如，对于一个正方形，在平面上连续旋转90度的集合是对称的集合，并产生大小为4的组。描述组的一种有用方法是通过所谓的表示，使用生成器和生成器之间的关系完全描述组。例如，所有整数的集合是由单个元素即整数1生成的无限群。如果一个组具有有限的表示（具有有限数量的生成器和有限数量的关系），那么就更容易理解，也更容易解决有关该组的问题。此外，如果一个群具有每个有限生成的子群都是有限呈现的这一更强的性质，则称为相干群。许多最容易理解的群是相干的：自由群、曲面群和3流形群都是相干的。该项目旨在了解群体何时是连贯的，并找到见证不连贯的子群体的几何指标。该项目通过研究看似行为良好的群体的几何野生子群来推进该领域，这些子群很难找到。该项目包括教育方面的工作，PI正在帮助组织会议和学校，以加强对下一代的培训。PI计划调查几类有趣的群体的一致性，包括预期一致的群体和预期不一致的群体。这个项目解决了关于非相干性和子群几何的基本和困难的问题。其中一个项目是一个著名猜想的关键步骤，即具有Sierpinski地毯边界的双曲群实际上是Kleinian。另一个是一个较新的猜想，即对两棵树的乘积起几何作用的群是不相干的。解决这些猜想将有助于理解一个群何时是3流形的基本群，以及重要群类的复杂子群结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A group is an algebraic object that is the collection symmetries of a geometrical object. For instance, for a square the set of successive rotations by 90 degrees in the plane is a collection of symmetries and yields a group of size 4. One useful way of describing a group is via what is known as a presentation, with generators and relations between the generators completely describing the group. For example, the collection of all integers is an infinite group that is generated by a single element, namely the integer 1. If a group has a finite presentation (with a finite number of generators and a finite number of relations) then it is easier to understand, and easier to solve problems about that group. In addition, a group is called coherent if it enjoys the stronger property that every finitely generated subgroup is finitely presented. Many of the best understood groups are coherent: free groups, surface groups, and 3-manifold groups are all coherent. This project aims to understand when groups are coherent and to find geometric indicators of subgroups that are witnesses to incoherence. This project advances the field by investigating geometrically wild subgroups of seemingly well-behaved groups, which can be difficult to find. The project includes work in education, and the PI is helping to organize conferences and schools that enhance the training of the next generation. The PI plans to investigate the coherence of several interesting classes of groups, including groups that are expected to be coherent and groups which are expected to be incoherent. This project addresses fundamental and difficult problems about incoherence and the geometry of subgroups. One of the projects is a key step to a well-known conjecture that hyperbolic groups with Sierpinski carpet boundary are virtually Kleinian. Another is a newer conjecture, that groups which act geometrically on the product of two trees are incoherent. Solving these conjectures will shed substantial light on understanding when a group is the fundamental group of a 3-manifold, and also on the intricate subgroup structure of important classes of groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.