Cayley graph

A graph representing a group relative to a generating set: group elements are vertices, and multiplication by generators determines edges. Its geometry depends on the chosen generators.

Group: A set equipped with an associative operation, an identity element, and an inverse for every element. Its elements supply the vertices, while its operation defines each edge.
Group presentation: A description of a group by generators and relations among them. Relations determine which generator words represent the same vertex.
Geometric group theory: The study of groups through geometric and topological methods, especially their actions on spaces. Cayley graphs are a central tool for comparing groups through geometry.
Directed graph: A graph whose edges have specified directions. Directed Cayley graphs can represent semigroup actions or one-way generator steps.
Generating set: A subset of a group whose finite products and inverses produce every group element. The chosen elements specify which steps connect neighboring vertices.
Free group: A group whose elements are reduced words in generators and their formal inverses, with no additional relations. Its Cayley graph is a tree, making it a basic model.
Growth rate of a group: The asymptotic count of group elements within increasing word-metric distance of the identity. Counting vertices in larger graph balls measures group growth.
Undirected graph: A graph whose edges have no orientation. Symmetric generating sets make generator steps reversible and the graph undirected.
Word metric: A distance on a group measuring the shortest expression of one element using a chosen generating set. Shortest paths in the graph give this algebraic distance.
Coset: A translate of a subgroup inside a group. Cosets can index vertices in related graphs built from group actions.

Linked from 15 pages