$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This is via the Lawrence–Krammer representation, proved to be faithful for all $n$ independently by Bigelow and Krammer. Thus, braid groups are linear.

My question is whether it is known that they are not linear over $\mathbb{Z}$. That is, is it known that, for large enough $n$, $B_n$ does not embed as a subgroup into any $\GL_m(\mathbb{Z})$? I haven't found any references to this being an open problem, so I'm wondering if there's just some established reason it can't happen.

soopen that very few people even talk about it in the literature. But I didn't know about this proposed Kontsevich representation, good to know. $\endgroup$1more comment