The Algebraic Uses of Geometric Group Theory

Abstract

Geometric group theory is an area of mathematics concerned with the study of finitely formed groups. When the 
groups in question are realized as geometric symmetries or continuous transformations of some spaces, this area 
of mathematics explores the relationships between the algebraic properties of these groups and the topological 
and geometric properties of the spaces on which they act. Early examples of this topic may be found in William 
Rowan Hamilton's 1856 icosian calculus, in which he examined the icosahedral symmetry group using the edge 
graph of the dodecahedral lattice. Walther von Dyck, a pupil of Felix Klein, first conducted systematic research 
in this field in the early 1880s. Combinatorial group theory, which mainly focused on examining the 
characteristics of discrete groups by examining group presentations, gave rise to geometric group theory. The 
topic of geometric group theory is now largely absorbing the study of combinatorial group theory. Furthermore, 
"geometric group theory" started to include the study of discrete groups using probabilistic, measure-theoretic, 
arithmetic, analytic, and other techniques that are not often used in combinatorial group theory.