Arithmetic group

Summary

An arithmetic group is a group of integer matrices cut out by polynomial equations, such as $\mathrm{SL}_n(\Z)$. A group that agrees with such a group up to finite index is also arithmetic. Arithmetic groups link number theory with the geometry of Lie groups. For a semisimple group, a theorem of Borel and Harish-Chandra says that the arithmetic subgroups are lattices in the real Lie group. The Margulis arithmeticity theorem shows that in higher rank all irreducible lattices arise this way.