Lattice in a Lie group

Summary

A lattice in a Lie group $G$ is a discrete subgroup $\Gamma$ such that the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure. It is the case of Lie groups of a lattice in a locally compact group. The lattices in $\R^n$ are the classical lattices of the geometry of numbers, and $\mathrm{SL}_n(\Z)$ is a lattice in $\mathrm{SL}_n(\R)$. Lattices link Lie groups with number theory, through arithmetic groups, and with geometry, as fundamental groups of finite-volume locally symmetric spaces.