## Some special actions

Consider the conjugate action $\rho$ of $GL(2,R)$ on $M(2,R)$: $\rho_A(M) = A MA^{-1}$.

1. This action $\rho$ factors through an action of $PGL(2,R)$.

2. There exists a 3D invariant subspace $E=\{M\in M(2,R): tr(M)=0\}$.

3. The determinant $\det M$ is an invariant quadratic form on $E$, and the signature of this form is $(-, - ,+)$.

Let $Q=x_1^2 + x_2^2 - x_3^2$ be a quadratic form on $R^3$, whose isometry group is $O(2,1)=\{A\in M(3,R): A^TgA=g\}$, where $g=\mbox{diag}\{1, 1, -1\}$.

This induces an injection $PGL(2,R) \subset O(2,1)$, and an identification between $PSL(2,R)$ and the connected component of $O(2,1)$.

The action $O(2,1)$ on $R^3$ passes on to the projective space $P^2$. The cone $C=Q^{-1}(0)$ is invariant, and separates $P^2$ into two domains: one of them is homeomorphic to a disk, which the other is a Mobius band. This induces an action of $PSL(2,R)$ on the disk.