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I need a reference with a proof that the inverse image of SL2(Z) in the universal covering of SL2(R) is the group of the trefoil knot (i.e. the Braid group B3 on 3 strands)

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    I believe you mean the fundamental group of the complement of the trefoil knot in $\Bbb S^3$, not that of the knot itself. – 2017-02-02
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    Yes, thank you, edited. – 2017-02-02
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    Do you have a link to an intuitive description of [$\widetilde{SL}_2(\mathbb{R})$](https://en.wikipedia.org/wiki/SL2(R)#Topology_and_universal_cover) the universal covering space of $SL_2(\mathbb{R})$ ? – 2017-02-02
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    No, sorry, I don't know much about covering spaces. The only thing I've found is [this answer](http://math.stackexchange.com/questions/381221/what-is-the-universal-cover-of-sl2-r) – 2017-02-02

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Reference: R. Hain: Lectures on Moduli Spaces of Elliptic Curves, in Transformation Groups and Moduli Spaces of Curves, Advanced Lectures in Mathematics, edited by Lizhen Ji, S.-T. Yau no. 16 (2010), 95–166, Higher Education Press, Beijing, arXiv:0812.1803, Corollary $8.3$.

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    I'll wait a while to see if someone has a reference which does not relie on moduli spaces. Then I'll accept the answer. Thank you. – 2017-02-02
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    In case you would like to have another reference, we can find one. Let me know what mathematical background you have, so that it is useful for you. – 2017-02-02
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    I've found this statement in example 1.5.2 of the book "Trees" by Jean-Pierre Serre. The chapter it belongs is about free products with amalgamation. So any proof related with it would be great (maybe one where I can see a sort of "concrete" visualization of what's going on). Other topics I'm familiar with (which may be useful in this situation) are braid groups, coxeter groups and standard topology. – 2017-02-02
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    Then the blog of [John Baetz](http://math.ucr.edu/home/baez/week233.html) might be useful. – 2017-02-02