Questions in and around Geometric Group Theory

这个项目的核心是问题，首先提出的格罗莫夫，分类的非线性生成组的（准等距）几何。 PI的建议遵循两条研究原则。 第一个方向是阐明问题的准等距分类各种家庭的群体。 第二个方向的研究，涉及应用知识从分析的准等距结构的一个家庭的群体，以阐明问题的不同性质：例如嵌入的三维流形组或功能分析性质的映射类groups.Quasi-isometric几何是一种考虑几何性质的空间是不变的“小”变化的距离函数。 在这个建议中，PI认为几何的各种自然家庭的群体和研究的问题，如：在何种程度上做几何性质决定代数的。 PI继续与Walter Neumann（巴纳德学院）一起研究的一个群体家族与三维流形有关-这些空间在低维拓扑和物理学中至关重要。

Central to this project is the question, first proposed by Gromov, of classifying finitely generated groups by their (quasi-isometric) geometry. The PI's proposal follows two principle lines of study. The first direction is to shed light on the question of quasi-isometrically classifying various families of groups. The second direction of study, involves applying knowledge gained from analysis of the quasi-isometric structure of a family of groups to illuminate questions of a different nature: for instance to embeddings of 3-dimensional manifold groups or to functional analytic properties of mapping class groups.Quasi-isometric geometry is a way of considering geometric properties of a space which are unchanged by a "small" change of the distance function. In this proposal the PI considers the geometry of various natural families of groups and studies question such as: to what extent do geometric properties determine algebraic ones. One family of groups which the PI continues to study with Walter Neumann (Barnard College) are those associated to 3-dimensional manifolds---these spaces are of crucial importance throughout low-dimensional topology and in physics as well.