Geometric Group Theory
Introduction
Geometric group theory is a field of mathematics that uses the principles of algebra and geometry to understand the structure of groups. It is a branch of group theory, which is a part of abstract algebra, and it uses tools from both topology and geometry to study groups.
History
Geometric group theory has its roots in the late 19th and early 20th centuries with the work of mathematicians such as Klein, Poincaré, and Dehn. However, it wasn't until the 1980s that the field began to take its current form, largely due to the work of Gromov.
Basic Concepts
Groups
In mathematics, a group is a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms.
Spaces
A topological space is a set of points, along with a set of neighbourhoods for each point, that satisfies a set of axioms relating points and neighbourhoods. In geometric group theory, groups are often studied via their actions on topological spaces.
Group Actions
A group action is a way of describing symmetries of objects using groups. In geometric group theory, group actions are often used to study the structure of groups.
Key Concepts in Geometric Group Theory
Cayley Graphs
A Cayley graph is a graph that encodes the structure of a group. It is named after the British mathematician Arthur Cayley.
Word Metric
The Word metric is a way of measuring distance in a group. It is defined in terms of the number of "steps" it takes to get from one element of the group to another.
Hyperbolic Groups
Hyperbolic groups are a class of groups that were introduced by Gromov. They have many interesting properties and are a major focus of study in geometric group theory.
Applications
Geometric group theory has applications in many areas of mathematics, including algebra, topology, and number theory. It has also found applications in computer science, particularly in the study of algorithms and complexity theory.