Aspects of Non-positively Curved Groups

群论是一个数学领域，它使用代数方法来抽象对称性的概念。一个正方形有8个不同的对称性，包括4个旋转和4个反射。与此相反，有些物体具有无限多的对称性，就像用正方形平铺欧几里得平面一样。对对称性群的研究已经应用于宇宙学、密码学或机器人学等不同的科学领域。该研究计划涉及无限对称群的研究。该程序是分类和理解这类群体结构的难题的一部分。研究人员将探索使用特定的组合概念的曲率拓扑复合物，以研究结构的离散群体对他们的作用。这涉及到使用复杂的技术属于该地区的代数拓扑，双曲几何，组合群论。该建议可以归类于几何群论的一般领域。这项研究计划的成果将提高我们对无限离散群的几何和代数性质之间的联系的理解，并将揭示一些几何群论中悬而未决的问题。该研究计划包含一个培训方面，概述了本科和研究生水平的研究项目。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.