Discrete group

Since topological groups are homogeneous, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete only if the singleton containing the identity is an open set.

A discrete group is the same thing as a zero-dimensional Lie group (uncountable discrete groups are not second-countable, so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself.

Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete.

A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.

Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian.

Other properties:

• every discrete group is totally disconnected
• every subgroup of a discrete group is discrete.
• every quotient of a discrete group is discrete.
• the product of a finite number of discrete groups is discrete.
• a discrete group is compact if and only if it is finite.
• every discrete group is locally compact.
• every discrete subgroup of a Hausdorff group is closed.
• every discrete subgroup of a compact Hausdorff group is finite.

• 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 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 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 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.

• crystallographic point group
• congruence subgroup
• arithmetic group
• geometric group theory
• computational group theory
• freely discontinuous
• free regular set

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