Grant: Discovery Grants Program - Individual

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.