Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$ Let's consider the pseudosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by
$$x^2+y^2-z^2=-R^2.$$
We know that the Lorentz group
$$O(1,2)=\{ A \in {\rm Mat}(3,\mathbb{R}): A^tGA=G \},$$
where $G={\rm diag}(-1,-1,1)$ leaves the pseudosphere invariant. Now we are interested in the following facts:

*

*How can we show that the orthochronous Lorentz group $O_+(1,2)=\{ A: a_{33}>0 \}$ is subgroup and, more important, maps upper cone to upper cone?


*What is the relation between groups $O_+(1,2)$ and $SL(2,\mathbb{R})$?
 A: For 1) @Vibert gives you the indications.
For 2) The group $0(1,2)$ - with signatures (+ - -) has 4 disjoint components which can be characterized by : 
$$M_1 = Diag (1, 1, 1)$$
$$M_2 = Diag (1, -1, -1)$$
$$M_3 = Diag (-1, 1, 1)$$
$$M_3 = Diag (- 1, -1, -1)$$
$S0(1,2)$ corresponds to matrix of determinant 1, so $S0(1,2)$ has 2 disjoint components ($M_1, M_2$)
$0^+(1,2)$ - which conserve the sign of the 1st coordinate -  has 2 disjoint components ($M_1, M_2$)
$S0^+(2,1)$ -  has 1 component ($M_1$)
$SL(2,\mathbb{R})$ is connected (so only 1  component), but it is not simply connected.
So, it is not possible to have a isomorphism between $SL(2,\mathbb{R})$ and $0^+(1,2)$ because the number of disjoint components is different.
We could think about an isomorphism between $SL(2,\mathbb{R})$ and $S0^+(1,2)$, but in fact the isomorphism is between $SL(2,\mathbb{R})$ and $Spin^+(1,2)$, while there is an isomorphism between $PSL(2,\mathbb{R})$ and $S0^+(1,2)$, see Wikipedia
Note that $SL(2,\mathbb{R})$, $SU(1,1)$, and $Sp(2,\mathbb{R})$ are isomorphic, see this question.
