Discrete subgroup

Summary

A discrete subgroup of a topological group is a subgroup whose elements are isolated from each other. In a Hausdorff topological group, discrete subgroups are closed. A discrete subgroup of a Lie group $G$ acts properly discontinuously on each quotient $G/K$ with $K$ compact. The resulting quotient spaces are central objects of geometry and number theory. Lattices, Fuchsian groups, Kleinian groups and crystallographic groups are all discrete subgroups.