On the representation theory of finitely generated groups

群是编码对称性的数学对象。整数是一个重要的群的例子，它编码的对称性，比如说，一排等距的汽车在一条笔直的高速公路上。线性组，这个建议的主要焦点，是那些组，在某种意义上类似于整数组的例子，编码的对称性的真实的生活对象。不是所有的群都是线性的，但是当你可以确定一个群是线性的时，这就打开了对这个群的全新理解（特别是，你可以应用线性代数的所有工具）。这个建议的目的是了解灰色区域，通过形成数字不变量和障碍，分层这个灰色区域，线性和非线性组分开。虽然这一建议将给予洞察力的理论群体，它也有应用到计算机科学和接口代数几何和几何群论。此外，拟议的工作将影响本科生和研究生在纽约城市学院（CCNY）通过辅导有关本提案的主题。这个建议的主要重点是从一个固定群到一组具有复杂条目的可逆矩阵的同态构造的复杂性。这一重点建立在以前的研究，其中PI探索了格罗莫夫的字增长和Lubotzky-Segal的子群增长的相互作用，通过开发量化剩余有限性的程序。这个程序给出了一个系统的方法来研究从一个固定群到任何有限群（也是线性的）的同态的复杂性。此外，该程序为发现新的特征的基本性质的群生成，包括幂零性，线性，算术性，甚至有限presentability铺平了道路。在继续追求量化剩余有限性的同时，PI将通过详细描述从固定群到半单线性代数群的同态的行为来增强他目前的研究计划（即，一个特殊的线性群）。特别是，PI试图确定何时自由群上的博雷尔定理推广到其他群。这一研究方向将影响对算术群体和词图行为的理解。算术群在几何学中处于中心地位，因为它们提供了流形的明确例子。词映射的研究从豪斯多夫-巴拿赫-塔斯基悖论的证明中找到了它的动机，并且与扩展图理论（理论计算机科学中的一个非常感兴趣的主题）交织在一起。虽然提出的项目是基于拓扑学，他们的特点是从代数几何，群论和数论的方法。

Groups are mathematical objects that encode symmetries. The integers are an important example of a group that encodes the symmetries of, say, a line of equally spaced cars on a straight highway. Linear groups, a main focus of this proposal, are those groups that, in a sense similar to the integer group example, encode symmetries of real life objects. Not all groups are linear, but when you can determine that a group is linear this opens up a whole new understanding of the group (in particular, you can apply all the tools from the subject of linear algebra). This proposal aims to understand the grey area that separates linear and nonlinear groups by forming numerical invariants and obstructions that stratify this gray area. While this proposal will give insight in the theory of groups, it also has applications to computer science and interfaces algebraic geometry and geometric group theory. Further, the proposed work will impact undergraduate and graduate students at the City College of New York (CCNY) through mentoring on topics related to this proposal. This proposal's primary focus is on the complexity of constructing homomorphisms from a fixed group to a group of invertible matrices with complex entries. This focus builds on previous research, where the PI explored the interaction of Gromov's word growth and Lubotzky-Segal's subgroup growth by developing the program of quantifying residual finiteness. This program gives a systematic approach for studying the complexity of constructing homomorphisms from a fixed group to any finite group (which is also linear). Further, this program paves way for the discovery of new characterizations of fundamental properties of finitely generated groups including nilpotency, linearity, arithmeticity, and even finite presentability. While continuing the pursuit of quantifying residual finiteness, the PI will augment his current research program by detailing the behavior of homomorphisms from a fixed group into a semisimple linear algebraic group (i.e., a special linear group). In particular, the PI seeks to determine when Borel's theorem on free groups generalizes to other groups. This research direction would influence the understanding of the behavior of arithmetic groups and word maps. Arithmetic groups are central in geometry because they supply explicit examples of manifolds. The study of word maps finds its motivation from the proof of the Hausdorff-Banach-Tarski paradox and is intertwined with the theory of expander graphs (a subject of great interest in theoretical computer science). While the projects proposed are grounded in topology, they feature methods from algebraic geometry, group theory, and number theory.