Residually Finite Braid Group In Braid Groups of Kassel, Turaev, it mentions that $\mathcal{B}_n$ is a residually finite group. The definition that they give as a residually finite group is a group $G$ such that for each $g\in G-\{e_G\}$ ($e_G$ the identity of $G$), there exists an homomorphism $f$ to a finite group $H$ such that $f(g)\neq e_H$. My question is:
How can I obtain the group $H$ for a given element $g\in \mathcal{B}_n$ and the homomorphism that fulfills this?
I hope you can help me. Nice Holidays.
 A: If the braid is not pure, then you can detect it by a homomrphism to the symmetric group. So this reduces to showing that the pure braid group $P_n$ is residually finite. $P_n$ is an iterated semi-direct product of free groups. (This is called combing the braid.) The result now follows because free groups are residually finite. https://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite
A: Another possibility is to embed $\mathcal{B}_n$ into the automorphism group $\mathrm{Aut}(\mathbb{F}_n)$ of the free group $\mathbb{F}_n$. Now, Baumslag gave a very short prove of the fact that, for any finitely generated residually finite group $G$, $\mathrm{Aut}(G)$ is also residually finite. The conclusion follows from the residual finiteness of finitely generated free groups (see for example the beautiful proof of Stallings in Topology of finite graphs), since a subgroup of a residually finite group is clearly residually finite itself.
For more details, see Basic results on braid groups and the references therein.
