# Lattice in a Lie group

A discrete subgroup $\Gamma$ of a Lie group $G$ such that $G/\Gamma$ has a finite volume relative to the (induced) $G$-invariant measure.

A lattice of dimension $n$ (or rank) $n$ in a vector space $V$ over $\mathbf R$ or $\mathbf C$ is a free Abelian subgroup in $V$ generated by $n$ linearly independent vectors over $\mathbf R$. A subgroup of the additive group of a finite-dimensional vector space $V$ over $\mathbf R$ is discrete if and only if it is a lattice [1].

#### References

 [1] S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977)