Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 1 / 2 / a

The Lie algebra of SL2R is the space of traceless real two-by-two matrices. For $A$ in this space, the Cayley-Hamilton theorem gives $A^2=cI$, where $c=-\det A$. Expanding the matrix exponential, if $c=s^2>0$ thenIf $c=0$, then $\exp A=I+A$ and its trace is $2$. If $c=-s^2<0$, thenIn every case $\mathrm{tr}(\exp A)\ge -2$. Buthas determinant one and trace $-5/2$. Thus $g\in S{L}_2(\mathbb{R})$ is not an exponential, and the exponential map is not surjective.