Constructing subgroups in hyperbolic groups

摘要奖：DMS 1405466，首席研究员：丹尼·卡尔加里群是将对称概念形式化的数学对象。它们可以是混合不同尺度的对称性(描述树木生长的递归对称性)，也可以描述大量类似物体的聚合行为(理想气体、有效市场)。我们永远无法完全理解群体--他们表现出任意复杂的现象，我们能问的最简单的问题也不能在“最坏的情况”下得到回答。但在“一般”情况下，情况就大大简化了。主要研究人员和他的合作者开发的技术允许人们构造具有很大控制力的一般群的子群，并证明这些子群具有任意数量的所需性质。该项目的目的是将这些技术扩展到更广泛的组类别，允许在构建中具有更大的灵活性，并在实际软件中实现这些理论工具。一个关键的想法是将拓扑学和几何学--也就是2维和3维的“双曲”几何--移植和构建到一般的组合世界中，在那里人们事先不会期望找到它们。这个项目的主要目的是开发工具来构造、证明和组织双曲群中的某些类拟凸子群，无论是在一般情况下(即随机群，在各种模型中)都是如此。我们关心的子群应该是我们所理解的群(即闭曲面的基本群或紧致非圆柱3-流形)，并且它们在环境群中的性质应该具有短证书(这是首先构造它们的关键)。主要的研究目的是在与不同合作者发展的技术的基础上，在各种类型的双曲群中构造这样的表面子群和非柱面3-流形子群，以期解决Gromov的表面子群问题，以及这个问题的各种推广和特化。开发的构建随机组中的对象的技术可以适用于一般的双曲线组，其中环境组的随机性代替了构建方法中的“自由度”。最后，作者期望在计算机上实现这些结构，证明它们作为理解特定类别组的工具的实用性，超出了它们所提供的理论见解。

AbstractAward: DMS 1405466, Principal Investigator: Danny CalegariGroups are the mathematical objects that formalize the idea of symmetry. These can be symmetries that mix different scales (recursive symmetries that describe the growth of trees), or that describe the aggregate behavior of vast numbers of similar objects (ideal gases, efficient markets). A complete understanding of groups is forever beyond our reach -- they exhibit phenomena of arbitrary complexity, and the simplest questions we can ask cannot be answered in the "worst case." But in the *generic* case the situation simplifies enormously. Techniques developed by the principal investigator with his collaborators allow one to construct subgroups of generic groups with a great deal of control, and to certify that these subgroups have any number of desired properties. The aim of this project is to extend these techniques to broader classes of groups, to allow more flexibility in the constructions, and to implement these theoretical tools in practical software. A key idea is to transplant and construct objects from topology and geometry -- namely *hyperbolic* geometry in dimension 2 and 3 -- into the generic combinatorial world, where a priori one would not expect to find them. The familiarity of these transplanted products provides the necessary leverage to break up the ambient group into understandable pieces.The main aim of this project is to develop tools to construct, to certify and to organize certain classes of quasiconvex subgroups in hyperbolic groups, both in the generic case (i.e. random groups, in various models) and in general. The subgroups we are concerned with should be groups that we understand (i.e. fundamental groups of closed surfaces or compact acylindrical 3-manifolds), and their properties in the ambient group should have short certificates (which are the key to their construction in the first place). The principal investigator aims to build on techniques developed with various collaborators to construct such surface subgroups and acylindrical 3-manifold subgroups in various classes of hyperbolic groups, with the view to attacking Gromov's surface subgroup question, and various generalizations and specializations of this question. Techniques developed to construct objects in random groups can be adapted to general hyperbolic groups, where "randomness" of the ambient group is substituted for "degrees of freedom" in the method of construction. Finally, the author expects to implement these constructions on computer, proving their practical utility as a tool to understand specific classes of groups, beyond the theoretical insights they provide.