Borel density theorem

Remarks

Furstenberg's proof for clause 1 goes as follows. View a $\Gamma$-invariant subspace $W$ of dimension $k$ as a point of the projective space $P$ of $\Lambda^k V$. The map $g\Gamma \mapsto \rho(g) W$ pushes the invariant probability measure on $G/\Gamma$ to a $G$-invariant probability measure on $P$. If $\rho(u)$ is unipotent, the points $\rho(u)^m x$ converge to fixed points of $\rho(u)$, so such a measure lives on these fixed points. As $G$ is generated by such elements $u$, the measure lives on $G$-fixed points, and this forces $W$ to be $G$-invariant.

Example: $\Gamma = \mathrm{SL}_2(\Z)$ in $G = \mathrm{SL}_2(\R)$, with the standard representation on $\R^2$. A line invariant under $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ must be $\R e_1$, and a line invariant under $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ must be $\R e_2$. So no line is invariant under $\Gamma$, as clause 1 predicts, since no line is invariant under $G$.

The hypothesis on compact factors is necessary. In the compact group $G = \mathrm{SO}(3)$, the trivial subgroup is a lattice, and every subspace of $\R^3$ is invariant under it. But the axis $\R e_3$ is not invariant under $\mathrm{SO}(3)$.

Consequences: the centraliser of $\Gamma$ in $G$ is the centre of $G$, and $\Gamma$ is not contained in any proper algebraic subgroup.

Going deeper

For a student

The Borel density theorem is the first step of many rigidity results. It lets one pass from statements about $\Gamma$ to statements about $G$, for instance in the Margulis superrigidity theorem, where Zariski density is part of the setup. It also shows that the adjoint representation of $G$ restricted to $\Gamma$ is irreducible when $G$ is simple, which is used for local rigidity of lattices.