I want to know about books, expository papers o lecture notes on the universal covering group $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ of the Lie group $\mathrm{SL}_2(\mathbb{R})$.
The Wikipedia page about $\mathrm{SL}_2(\mathbb{R})$ contains some interesting facts about this group, but gives no reference for them. For example I'm interested in:
- Proofs that $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ has no faithful finite-dimensional representations.
- A better understanding of the realization of the modular group $\mathrm{SL}_2(\mathbb{Z})$ as a quotient of the Artin braid group $B_3$, which is its pre-image under the covering projection $\widetilde{\mathrm{SL}_2(\mathbb{R})}\to \mathrm{SL}_2(\mathbb{R})$.
- Facts than can be deduced from the fact that we have covering projections $\widetilde{\mathrm{SL}_2(\mathbb{R})}\to \mathrm{Mp}_2(\mathbb{R})\to \mathrm{SL}_2(\mathbb{R})$, where $\mathrm{Mp}_2(\mathbb{R})$ is the metaplectic group.
- Possible realizations of $\widetilde{\mathrm{SL}_2(\mathbb{R})}$. For example the metaplectic group $\mathrm{Mp}_2(\mathbb{R})$ can be realized as the group of pairs $(g,f)$ where $g\in\mathrm{SL}_2(\mathbb{R})$ and $f:\mathcal{H}\to \mathbb{C}$ is a homolomorphic function on the upper-half plane $\mathcal{H}$ with the property that $f$ is a square root of the automorphy factor $j(a,\tau)=c\tau+d$, where $g=\begin{pmatrix} a & b\\ c &d \end{pmatrix}$.
I doubt this will answer all the parts of your question, but the following paper on Geometries of 3-manifolds by Peter Scott has a nice description of $\tilde SL_2(\mathbb{R})$.
Algebraically, it can be thought of as a central extension of $SL_2(\mathbb{R})$ and geometrically, it is the universal cover of the unit tangent bundle of $\mathbb{H}^2$.