Lattice in a locally compact group

Summary

A lattice in a locally compact group $G$ is a discrete subgroup $\Gamma$ of finite covolume. That is, the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure. It is cocompact, or uniform, when $G/\Gamma$ is compact. Lattices are discrete approximations of $G$; the integer lattice $\Z^n$ in $\R^n$ and the modular group $SL_2(\Z)$ in $SL_2(\R)$ are the guiding examples. They link group theory with number theory, geometry and ergodic theory.