I was wondering about perfect quotients of the Braid groups $ B_n $. Certainly $ B_1=1 $ and $ B_2=\mathbb{Z} $ have finitely many perfect quotients. For $ B_3 $, $$ 1 \to Z(B_3)=\mathbb{Z} \to B_3 \to PSL(2,\mathbb{Z}) \to 1 $$ I was wondering about perfect quotients of $ B_3 $, if they have a nice description, and , at even more basic level, if there are finitely many of them.
This led me to the general question:
Does a finitely generated group have finitely many perfect quotients?
Obviously $ \mathbb{Z} $ is a finitely generated group with infinitely many simple quotients $ \mathbb{Z}/n\mathbb{Z} $ but these are not perfect.
No a finitely generated group does not always have finitely many perfect quotients.
In fact, every Braid group has an unbounded number of finite perfect quotients.
Ian Agol notes here
https://mathoverflow.net/a/449481/387190
that the paper A’Campo, Norbert, Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54, 318-327 (1979). ZBL0441.32004 MR0535062
shows that every braid group $ B_n $ has quotient $ Sp(m,\mathbb{Z}) $ for some $ m $. And thus, for that $ m $, every finite perfect group $ Sp(m,p) $ for $ p $ prime is a quotient of $ B_n $. Thus every $ B_n $ has infinitely many finite perfect quotients.