Reference request: Universal central extension of $\operatorname{PSl}_2(\mathbb Z)$ is the braid group $\mathcal B_3$.

Question

$\DeclareMathOperator{\PSl}{PSl}$ According to Wikipedia, the universal central extension of $\PSl_2(\mathbb Z)$ is given by the braid group $\mathcal B_3$ on three strands, \begin{equation} \label{extension} \tag{1} 0 \to c \mathbb Z \to \mathcal B_3 \xrightarrow{\rho} \PSl_2(\mathbb Z) \to 1.\end{equation} What is a good reference for this?

Let's fix some notation: The braid group $\mathcal B_3$ is generated by the elementary twists $A$ and $B$, subject to the braid relation \begin{equation} \label{braid-relation} \tag{2} ABA = BAB.\end{equation} The map $\rho$ is given by $$A \mapsto \begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}, \quad B \mapsto \begin{bmatrix} 1 & 0 \\ 1 & 1\end{bmatrix}.$$ The element $c \in \mathbb B_3$ is $c = (ABA)^2 = (AB)^3$. So far I have checked:

1. The map $\rho$ is well-defined, i.e. it preserves the braid relation \eqref{braid-relation}.
2. The map $\rho$ is surjective, it hits the typical generators of $\PSl_2(\mathbb Z)$ corresponding to the Möbius transformations $z \mapsto - \frac 1 z$ (this is $\rho(A^{-1})$) and $z \mapsto z + 1$ (this is $\rho(ABA)$).
3. The element $c$ is indeed central (commutes with $A$ and $B$) and is in the kernel of $\rho$.

What I'm still missing is:

1. The kernel of $\rho$ is generated by $c$.
2. The extension $\eqref{extension}$ is universal.

What is a good reference for this? Or is that easily seen?

Answer 1

I found an article on arXiv where the author studies the relationship between the triaxial group and $\operatorname{PSl}_2(\mathbb Z)$, but the article does not directly state that $\operatorname{PSl}_2(\mathbb Z)$ is a universal central extension of the triaxial group​ 1

Regarding the first point, it can be noted that the kernel of a surjective homomorphism can be finitely generated if the original group is finitely generated and the target group is finitely represented 2. This may help you prove that the kernel $\rho$ is generated by the $c$ element.

The universal central extension is defined in such a way that for any other central extension there are unique homomorphisms that ensure the commutativity of the corresponding diagram​ 3​. This may give some insight into what "an extension is generic" means.