Margulis normal subgroup theorem

Remarks

The proof has two halves, for a non-central $N$. One half shows that $\Gamma/N$ is amenable, by an analysis of the measurable quotients of the boundary $G/P$, with $P$ a minimal parabolic subgroup. The other half shows that $\Gamma/N$ has property (T) when $G$ has it; when $G$ does not, a different argument replaces this half. A discrete group that is amenable and has property (T) is finite.

Example: $\Gamma = \mathrm{SL}_3(\Z)$. The centre of $\mathrm{SL}_3(\R)$ is trivial, so every nontrivial normal subgroup of $\mathrm{SL}_3(\Z)$ has finite index. The congruence subgroups, the kernels of reduction modulo $m$, are such normal subgroups.

The rank condition is necessary. The principal congruence subgroup $\Gamma(3)$ of $\mathrm{SL}_2(\Z)$ is normal, of finite index and free of rank at least $2$. Its commutator subgroup $N$ is characteristic in $\Gamma(3)$, so it is normal in $\mathrm{SL}_2(\Z)$. But $N$ is infinite and has infinite index, since $\Gamma(3)/N$ is free abelian of positive rank.

Going deeper

For a student

The theorem shows that higher rank lattices are just infinite up to the centre. It is a strong contrast to lattices in $\mathrm{SL}_2(\R)$, which are often virtually free and have a huge supply of normal subgroups. It works together with the Margulis superrigidity theorem and the Margulis arithmeticity theorem to describe these lattices completely.