Aspects of Non-positively Curved Groups

群论是使用代数方法抽象对称性概念的数学领域。正方形有8个不同的对称，由4个旋转和4个反射组成。相反，存在具有无限多个对称性的物体，如欧几里得平面的平方平铺。对对称性群的研究已经应用于不同的科学领域，如宇宙学、密码学或机器人学。这项研究计划涉及对无限群对称的研究。该计划是对这类群体的结构进行分类和理解这一难题的一部分。研究人员将探索在拓扑复形上使用特定的组合曲率概念来研究作用在其上的离散群的结构。这涉及到使用属于代数拓扑学、双曲几何和组合群论领域的复杂技术。这个提议可以归类为几何群论的一般领域。这一研究计划的结果将加深我们对无限离散群的几何性质和代数性质之间的联系的理解，并将对几何群论中的一些悬而未决的问题有所启发。研究计划包含一个培训方面，其中概述了本科生和研究生层面的研究项目。这些项目旨在培养低维拓扑学和几何群论领域的高素质人才。

Group theory is the area of mathematics that uses algebraic methods to abstract the idea of symmetry. A square has 8 distinct symmetries consisting of 4 rotations and 4 reflections. In contrast, there are objects with infinitely many symmetries as the tiling of the Euclidean plane by squares. The study of groups of symmetries have had applications to diverse areas of science like cosmology, cryptography, or robotics. The research proposal deals with the study of infinite groups of symmetries. The program is part of the difficult problem of classifying and understanding the structure of this class of groups. The investigator will explore the use of specific combinatorial notions of curvature on topological complexes to study the structure of discrete groups acting on them. This involves the use of sophisticated techniques belonging to the areas of algebraic topology, hyperbolic geometry, and combinatorial group theory. The proposal can be classified in the general area of geometric group theory. The outcome of this research program will improve our understanding of the connection between geometric and algebraic properties of infinite discrete groups, and will shed some light on outstanding open problems in geometric group theory. The research program contains a training aspect which outlines research projects at the undergraduate and postgraduate levels. These projects aim to produce highly qualified personnel in the areas of low dimensional topology and geometric group theory.