Discrete group

In mathematics, discrete subgroups of topological groups play an important role in topology, differential geometry and the theory of Lie groups.

Definition

Let be a topological group. A subgroup is called discrete if the induced subspace topology is the discrete topology, ie all elements are isolated: in a sufficiently small neighborhood of any element are no other elements of.

A representation of an ( abstract ) group is called discrete if the image is a discrete subgroup of.

Examples

• Is a discrete subgroup
• Is a discrete subgroup
• Is not a discrete subgroup
• Is a discrete subgroup

Properties

A discrete subgroup of a Hausdorff topological group is always complete.

Grid

Let be a locally compact - compact topological group, the projection and the (unique up to a constant factor) Hair measure. For a discrete subgroup Hair measure that produces a well-defined measure on as follows: for any quantity we define.

A lattice is a discrete subgroup for which there is a fundamental domain finite volume, or equivalently: for the quotient space finite volume (with respect to the hair dimension ) has.

The grid is called uniform or kokompakt if is compact.