Dirichletdomain

A Dirichlet domain, named after mathematician Johann Peter Gustav Lejeune Dirichlet, is a fundamental concept in the study of geometric structures, especially within the fields of hyperbolic, Euclidean, and spherical geometries. It is a specific type of fundamental domain associated with the action of a discrete group of isometries on a geometric space.

Formally, given a discrete group G acting on a metric space X, a Dirichlet domain centered at a point p in X is the set of points in X that are closer to p than to any of its images under the group action. In mathematical terms, it consists of all points x such that the distance from x to p is less than or equal to the distance from x to g(p) for every g in G, with equality on the boundary.

Dirichlet domains serve as a tool for understanding the tessellation of spaces under group actions. They are particularly useful in the study of tessellations of hyperbolic space, as they provide a convex polyhedral region that forms a representative 'tile' of the space with respect to the group. These domains are instrumental in the classification of discrete groups and their related tessellations, with applications spanning geometric topology, group theory, and complex analysis.

Constructing a Dirichlet domain involves selecting a base point and determining the perpendicular bisectors of lines connecting the base point to its images under the group transformations. The resulting intersection of half-spaces delineates the Dirichlet domain, which encapsulates the symmetry and structure of the group acting on the space.