Kazhdan–Margulis theorem

Remarks

The proof starts from a Zassenhaus neighbourhood: the elements of $\Gamma$ close to $e$ generate a nilpotent group. One then conjugates by suitable elements of $G$ that expand this nilpotent part, which is possible because $G$ has no compact factors.

Example: $G = \mathrm{SL}_2(\R)$ and $\Gamma = \mathrm{SL}_2(\Z)$. Conjugating the element $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ by $\operatorname{diag}(t, t^{-1})$ gives $\begin{pmatrix} 1 & t^2 \\ 0 & 1 \end{pmatrix}$, which is close to $e$ for small $t$. So some conjugates of $\Gamma$ come close to $e$, and the theorem only asserts that some conjugate stays away.

The hypothesis on compact factors is necessary. In the compact semisimple group $\mathrm{SO}(3)$, let $\Gamma_n$ be the finite cyclic group of rotations about one axis by multiples of $2\pi/n$. Every conjugate of $\Gamma_n$ contains a rotation by the angle $2\pi/n$. So for a given $U$, every conjugate of $\Gamma_n$ meets $U \setminus \set{e}$ once $n$ is large.

Going deeper

For a student

The main consequence concerns lattices. The volume of $G/\Gamma$ is at least a constant $c(G) > 0$ that does not depend on the lattice $\Gamma$. Indeed a small ball around $e$ injects into $G/(g\Gamma g^{-1})$, and $G/\Gamma$ and $G/(g\Gamma g^{-1})$ have the same volume. For quotients of symmetric spaces, this gives a uniform lower bound for the volume of locally symmetric orbifolds of a given type.