Discrete subgroup

Let $G$ be a Lie group .

Definition

A subgroup $\Gamma\le G$ is discrete if it is a discrete subset in the subspace topology (equivalently, every $\gamma\in\Gamma$ is isolated in $G$).

Basic Lie-theoretic consequences

• Any discrete subgroup is automatically closed; hence by the Closed Subgroup Theorem it is an embedded Lie subgroup .
• The Lie algebra of a discrete subgroup is trivial: $\mathrm{Lie}(\Gamma)=0,$ because there are no nontrivial smooth curves in $\Gamma$ through the identity.

If $\Gamma$ is also a normal subgroup, then the quotient $G/\Gamma$ is a Lie group (see quotient Lie group ), and the projection $G\to G/\Gamma$ is a covering Lie group map precisely when $\Gamma$ is discrete and acts freely by left translations.

Examples

• $\mathbb Z^n$ is a discrete subgroup of $\mathbb R^n$ (additively).
• The subgroup $\{\pm I\}$ is discrete in $SU(2)$ and is the kernel of the standard covering $SU(2)\to SO(3)$ (compare $SO(3)$ ).

Context. Discrete subgroups appear as “global” corrections to Lie-algebraic data: different discrete central quotients of a simply connected group yield different Lie groups with the same Lie algebra.