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.

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