The braid group $B_3$ has the property that its central quotient (i.e., $B_3 / Z(B_3)$) is isomorphic to the modular group $\mathrm{PSL}(2,\mathbb{Z})$. The modular group is known to be isomorphic to the free product of $(\mathbb{Z}/2\mathbb{Z}) \ast (\mathbb{Z}/3\mathbb{Z})$.
I'm curious about the extent to which this is true for braid groups $B_n$ for $n > 3$.
Question 1: For which $n > 3$ is it true that there exists two non-trivial groups $G$ and $H$ such that $B_n/Z(B_n)$ is isomorphic to $G \ast H$?
I'm also curious about a weaker version of this statement where we allow amalgamation over some shared finite subgroup.
Question 2: For $n > 3$ is it true that there are groups $G$ and $H$ with common finite proper subgroup $A$ such that $B_n/Z(B_n)$ is isomorphic to $G \ast_A H$?
