Lattice Points - in Lie Groups

In Lie Groups

More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant—the definition is independent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is not necessary, as is shown by the case of the modular group in SL₂(R), which is a lattice but where the quotient isn't compact (it has cusps). There are general results stating the existence of lattices in Lie groups.

A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.