Are braid groups known to not be linear over $\mathbb{Z}$?

$$\DeclareMathOperator\GL{GL}$$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.

• I guess it's an open question (I wondered about it long ago), for $n\ge 4$. I don't remember there's a known representation that would be a candidate to be faithful.
– YCor
Commented Feb 5, 2022 at 13:22
• Braid groups are subgroups of mapping class groups. Kontsevich gave a candidate representation of mapping class groups into $GL_m(\mathbb Z)$, with $m$ is about $g^g$. Commented Feb 5, 2022 at 14:45
• Yeah, I guess it's just open, and perhaps so open that very few people even talk about it in the literature. But I didn't know about this proposed Kontsevich representation, good to know. Commented Feb 6, 2022 at 18:28
• Is it known if $B_n$ is not virtually special? I think there was a question whether it is even CAT(0) and I do not know if it is resolved. Commented Feb 7, 2022 at 2:57
• see this question, answers and comments mathoverflow.net/questions/42836/… Commented Feb 7, 2022 at 3:12