Representations of 3-manifold groups

三维空间的数学研究可以追溯到庞加莱的工作（e。由于经典物理学将我们的宇宙描述为三维空间，因此三维空间的分类是一项重要的数学奋进，因为它可能对我们宇宙的整体结构产生影响。三维空间中最广泛和最有趣的一类是“双曲流形”，它由两个复矩阵的离散群描述。这些组可以表现为PI计划探索的高维矩阵。三维空间研究的重要性可能源于对空间上的“束”的研究，其将矢量数据附加到三维空间中的每个点。PI计划研究这种束的模量，包括尺寸和组件数量（结构可以相互变形）。该项目计划利用PI和其他人在理解三维双曲空间方面的最新进展。三维空间及其不变量的算法分类可能会产生分歧。本项目提出了关于三维流形群表示的几种研究途径。PI将研究双曲3-流形群的有限完备化。特别地，PI将探索关于3-流形的哪些信息可以从其基本群的有限向量中提取，包括双曲体积和秩。PI建议研究忠实不可约表示空间的维度，因为表示的维度增加。PI计划使用直角Artin群和嵌入3-流形来研究这个问题。一个相关的问题是研究双曲3流形群的n维表示的体积数随n的增加而增长。最后，PI将研究双曲拉紧缝合三维流形的扭同调。特别地，PI希望找到当它们是扭曲SL（2，C）-同调产物时的标准，其动机是确定扭曲SL（2，C）亚历山大多项式何时检测塞弗特亏格（或更一般地瑟斯顿范数）的问题。

The mathematical study of 3-dimensional spaces goes back to the work of Poincar(e. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. One of the most broad and interesting classes of 3-dimensional spaces are "hyperbolic manifolds", which are described by discrete groups of two by two complex matrices. These groups can have manifestations as higher-dimensional matrices, which the PI plans to explore. The importance for the study of 3-dimensional spaces may stem from the study of "bundles" over the spaces, which attach vector data to each point in the 3-dimensional space. The PI plans to investigate the moduli of such bundles, both in terms of dimension, and number of components (which structures can be deformed to each other). The project plans to take advantage of recent advances in the understanding of 3-dimensional hyperbolic spaces discovered by the PI and others. There may be ramifications for the algorithmic classification of 3-dimensional spaces and their invariants.This project proposes several avenues of study regarding representations of 3-manifold groups. The PI will investigate the profinite completions of hyperbolic 3-manifold groups. In particular, the PI will explore which information about a 3-manifold may be extracted from the finite quotients of its fundamental group, including hyperbolic volume and rank. The PI proposes to investigate the dimension of the space of faithful irreducible representations as the dimension of the representations increases. The PI plans to use right-angled Artin groups and embeddings of 3-manifolds into them to investigate this question. A related question is to investigate the growth of the number of volumes of n-dimensional representations of hyperbolic 3-manifold groups as n increases. Finally, the PI will investigate twisted-homology of hyperbolic taut sutured 3-manifolds. In particular, the PI would like to find criteria for when they are twisted SL(2,C)-homology products, motivated by the problem of determining when twisted SL(2,C) Alexander polynomials detect the Seifert genus (or more generally Thurston norm).