Artin groups are a class of groups defined by generators and relations, which generalize the braid groups and are associated with Coxeter groups. They play a significant role in geometric group theory and have applications in topology, particularly in the study of configuration spaces and hyperplane arrangements.

Artin Groups

Class of groups

Generators and relations are fundamental tools in group theory used to describe algebraic structures. Generators provide a set of elements from which the entire group can be formed, while relations are equations that these generators satisfy, defining the group's structure completely.

Generators And Relations

Braid groups are algebraic structures that capture the idea of braiding strands, with applications in topology, algebra, and mathematical physics. They are defined by generators and relations, where each generator represents a basic twist between two adjacent strands, and the relations capture the fundamental properties of these twists.

Braid Groups

Coxeter groups are abstract groups that can be represented by reflections across hyperplanes in a Euclidean space, and they play a fundamental role in the study of symmetries and geometric structures. They are defined by a set of generators subject to relations that can be encoded in a Coxeter matrix or diagram, allowing for the classification and analysis of symmetrical properties in various dimensions.

Coxeter Groups

Geometric group theory studies groups by exploring the geometric properties of spaces on which these groups act, often using techniques from topology and geometry. This approach provides deep insights into the algebraic structure of groups by examining their actions on metric spaces, particularly through the lens of Cayley graphs and hyperbolic spaces.

Geometric Group Theory

Applications in topology

Study of configuration spaces

A hyperplane arrangement is a finite collection of hyperplanes in a vector space, which provides a framework for studying various algebraic, geometric, and combinatorial properties. These arrangements are crucial in understanding the topology of the complement space, the structure of associated algebraic objects, and their applications in areas like optimization and machine learning.

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