Geometric Group Theory And Its Algebraic Applications

Abstract

The study of finitely generated groups is the focus of the branch of mathematics known as geometric group theory. This branch of mathematics investigates the connections between the algebraic properties of such groups and the topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). This field was first systematically studied by Walther von Dyck, a student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedral lattice. Geometric group theory evolved from combinatorial group theory, which primarily focused on the study of the properties of discrete groups through the analysis of group presentations At the moment, the field of combinatorial group theory is being mainly absorbed by the field of geometric group theory. In addition, the study of discrete groups by probabilistic, measure-theoretic, arithmetic, analytic, and other methods that fall outside of the typical armory of combinatorial group theory began to be included under the umbrella of "geometric group theory."