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This is borrowed from Clifford Taubes’s *differential geometry*:

The group $SL(2,\mathbb{R})$ is diffeomorphic to $\mathbb{S}^{1}\times \mathbb{R}^{2}$. This can be seen by using a linear change of coordinates on $M(2,\mathbb{R})$ that writes the entires in terms of $(x,y,u,v)$ as follows $$M_{1,1}=x-u, M_{22}=x+u,M_{12}=v-y,M_{21}=v+y$$

The condition $\det(M)=1$ now says that $x^{2}+y^{2}=1+u^{2}+v^{2}$. This understood, the diffeomorphism from $\mathbb{S}^{1}\times \mathbb{R}^{2}$ to $SL(2,\mathbb{R})$ sends a triple $(\theta,a,b)$ to the matrix determined by $$x=(1+a^{2}+b^{2})^{1/2}\cos[\theta],y=(1+a^{2}+b^{2})^{1/2}\sin[\theta],u=a,v=b$$ Here $\theta\in [0,2\pi]$ is the angular coordinate for $\mathbb{S}^{1}$.

And it should not be difficult for you to see the universal cover of this space is $\mathbb{R}^{3}$.

The typo in the solution was corrected by Taubes’ email.

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