Lattice in a Lie group

Remarks

When such a measure exists, it is unique up to a positive factor and is induced by a Haar measure $\mu$ of $G$. The total mass of $G/\Gamma$ for the measure induced by $\mu$ is the covolume of $\Gamma$. A Lie group that contains a lattice is unimodular. Example: $\Z^n$ is a uniform lattice in $\R^n$, because $\R^n/\Z^n$ is a compact torus whose Lebesgue measure is finite and invariant under translations. Example: $\mathrm{SL}_2(\Z)$ is a non-uniform lattice in $\mathrm{SL}_2(\R)$; the modular surface has finite hyperbolic area $\pi/3$ but is not compact. Non-example: $\Z \times \set{0}$ is discrete in $\R^2$, but $\R^2/(\Z \times \set{0})$ is an infinite cylinder, and any invariant measure is a multiple of its infinite area measure. Non-example: the subgroup $\Z + \sqrt{2}\,\Z$ of $\R$ is dense, not discrete, so it is not a lattice.