Editing Discrete group (section)

==Examples==
* [[Frieze group]]s and [[wallpaper group]]s are discrete subgroups of the [[isometry group]] of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not.
* A [[crystallographic group]] usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. Sometimes, however, a [[crystallographic group]] can be a cocompact discrete subgroup of a nilpotent or [[solvable Lie group]].
* Every [[triangle group]] ''T'' is a discrete subgroup of the isometry group of the sphere (when ''T'' is finite), the Euclidean plane (when ''T'' has a '''Z'''&nbsp;+&nbsp;'''Z''' subgroup of finite [[Index of a subgroup|index]]), or the [[Hyperbolic space|hyperbolic plane]]. 
* [[Fuchsian group]]s are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. 
** A Fuchsian group that preserves orientation and acts on the upper half-plane model of the hyperbolic plane is a discrete subgroup of the Lie group PSL(2,'''R'''), the group of orientation preserving isometries of the [[upper half-plane]] model of the hyperbolic plane.
** A Fuchsian group is sometimes considered as a special case of a [[Kleinian group]], by embedding the hyperbolic plane isometrically into three-dimensional hyperbolic space and extending the group action on the plane to the whole space.
** The [[modular group]] PSL(2,'''Z''') is thought of as a discrete subgroup of PSL(2,'''R'''). The modular group is a lattice in PSL(2,'''R'''), but it is not cocompact.
* [[Kleinian group]]s are, by definition, discrete subgroups of the isometry group of [[hyperbolic 3-space]]. These include [[quasi-Fuchsian group]]s.
** A Kleinian group that preserves orientation and acts on the upper half space model of hyperbolic 3-space is a discrete subgroup of the Lie group PSL(2,'''C'''), the group of orientation preserving isometries of the [[upper half-space]] model of hyperbolic 3-space.
* A [[lattice (discrete subgroup)|lattice]] in a [[Lie group]] is a discrete subgroup such that the [[Haar measure]] of the quotient space is finite.