Discrete group
From Wikipedia, the free encyclopedia
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not.
Since topological groups are homogeneous, one need only look at a single point to determine if the group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is a clopen set.
Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group. Hence, there is an isomorphism between the categories of groups and of discrete groups and indeed, discrete groups can generally be identified with the underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory.
If G is a finite or countably infinite group, then the discrete topology suffices to make it a zero-dimensional Lie group. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.
There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups.
A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.
[edit] Examples
- Frieze groups and wallpaper groups are discrete subgroups of the isometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not.
- A space group is a discrete subgroup of the isometry group of Euclidean space of some dimension.
- 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 + Z subgroup of finite index), or the hyperbolic plane.
- Fuchsian groups 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 is PSL(2,Z), 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 groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. These include quasi-Fuchsian groups.
- 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 in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite.
