In mathematics, specifically in the fields of geometry and group theory, a **fundamental domain** is a concept used to describe a specific subset of a space that can be used to represent an entire space under the action of a group. Here are some key points to understand about fundamental domains: 1. **Definition**: A fundamental domain for a group action on a space is a region that contains exactly one representative of each orbit of the action.
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For a group action, a fundamental domain selects one representative from each orbit. Geometric fundamental domains are often taken with their boundary included, allowing boundary overlaps while their interiors remain disjoint under the action. For translations of the real line by the integers, is an exact representative set and is its closed geometric version. A lattice similarly gives a half-open fundamental parallelepiped.