Margulis arithmeticity theorem

Remarks

The proof rests on the Margulis superrigidity theorem. Superrigidity is applied to $\Gamma$ in real, complex and $p$-adic groups. It shows that, in suitable coordinates, the matrix entries of $\mathrm{Ad}(\Gamma)$ lie in a number field and have bounded denominators. From this one builds the $\Q$-group $H$.

Example: $\mathrm{SL}_n(\R)$ with $n \ge 3$ has real rank $n - 1 \ge 2$. So every lattice in it is commensurable, up to conjugation, with a group built from an algebraic group over $\Q$, such as $\mathrm{SL}_n(\Z)$.

The rank condition cannot be dropped completely. Gromov and Piatetski-Shapiro constructed non-arithmetic lattices in $\mathrm{SO}(n, 1)$ for every $n \ge 2$. Mostow, and Deligne and Mostow, found non-arithmetic lattices in $\mathrm{SU}(2, 1)$ and $\mathrm{SU}(3, 1)$. In real rank $1$, the groups $\mathrm{Sp}(n, 1)$ and the rank one form of $F_4$ still have only arithmetic lattices, by Corlette and Gromov–Schoen.

Going deeper

For a student

Together with the Borel–Harish-Chandra theorem, this gives a complete description of irreducible lattices in higher rank. It turns questions about lattices into questions about algebraic groups over number fields. In rank one, arithmeticity can fail, and hyperbolic manifolds give many non-arithmetic examples.