The quotient of $ by the full twist squared has no proper quotient $\mathrm{SL}(2,\mathbb Z)$

The trefoil quotient of $has no proper quotient isomorphic to$\mathrm{SL}(2,\mathbb Z)$.

Let $ be Artin's braid group on three strands and let

\Delta^4 = (\sigma_1\sigma_2)^6

be the full twist squared. Write

Q = B_3 \big/ \langle!\langle \Delta^4 \rangle!\rangle

for the quotient by its normal closure. The claim is that every surjective homomorphism $\varphi : Q \twoheadrightarrow \mathrm{SL}(2,\mathbb Z)$ is injective, i.e. $ has no proper quotient isomorphic to the modular group.

The reason is structural. The images = \begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}$and = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$ of the standard generators generate $\mathrm{SL}(2,\mathbb Z)$, so $maps onto the modular group; and the quotient$ is generated by the two classes of $\sigma_0\sigma_1$ and $\sigma_0^2\sigma_1$, which satisfy exactly the amalgam relations

s^4 = 1, \qquad u^6 = 1, \qquad s^2 = u^3

with = \sigma_0^2\sigma_1$, = \sigma_0\sigma_1$ in $. Hence$ is the amalgamated product *_{C_2} C_6$, the standard presentation

\mathrm{SL}(2,\mathbb Z) ;=; \langle S, R \mid S^4 = 1,; R^3 = S^2 \rangle

of the modular group, with \mapsto \begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}$and \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 1\end{pmatrix}$. So \cong \mathrm{SL}(2,\mathbb Z)$.

Finally, $\mathrm{SL}(2,\mathbb Z)$ is finitely generated and residually finite (it is linear over $\mathbb Z$, and reduction of matrix entries modulo a prime bigger than the entries of - 1$separates any \neq 1$), and by Mal'cev's theorem a finitely generated residually finite group is Hopfian: every surjective endomorphism is injective. Composing a surjection $\varphi : Q \twoheadrightarrow \mathrm{SL}(2,\mathbb Z)$ with an isomorphism $\mathrm{SL}(2,\mathbb Z) \cong Q$ yields a surjective endomorphism of $, which is therefore injective, and so$\varphi$ itself is injective.

This is the group-theoretic core of the description of the kernel of the = -1$specialization of the reduced Burau representation: it is what upgrades the surjection$\psi : Q \twoheadrightarrow \mathrm{SL}(2,\mathbb Z)$induced by that specialization to an isomorphism, so that the kernel of the specialization is exactly the normal closure of$\Delta^4$.