# Is the word problem in the braid group quotient $B_{n}/N$ solvable where $N$ is the normal subgroup generated by conjugates of $\sigma_{i}^{2r}$?

Let $r\geq 2$. Let $N$ be the normal subgroup of $B_{n}$ generated by conjugates of $\sigma_{i}^{2r}$. Then is the word problem in the quotient group $B_{n}/N$ solvable (in polynomial time)? Furthermore, does there exist a normal form for the elements in $B_{n}/N$? I am mostly interested in this group where $r=3$, so $N$ is generated by the conjugates of $\sigma_{i}^{6}$.

• "normal subgroup generated by $x$" = "subgroup generated by conjugates of $x$" ("normal subgroup generated by conjugates of $x$" is redundant) – YCor Nov 22 '16 at 16:33
• This is known for small values of n and some ranges of $r$ but the general case is wide open as far as I know. My personal guess is that these groups are automatic for large (or even all) $r$. Derek Holt has software for verifying automaticity of groups, maybe you should ask him. – Misha Nov 22 '16 at 17:03
• I have tried these examples but without success. The braid groups are in any case difficult for my programs because you have to use the full set of $n!-1$ Garside generators to get a geodesic automatic structure. – Derek Holt Nov 22 '16 at 19:14
• Misha. Can you give a reference for your claim that it is known for certain values? – Joseph Van Name Nov 23 '16 at 0:01
• It will take a bit of work on my part. The examples I know stem from complex hyperbolic manifolds of small complex dimension and their ramified covers. – Misha Nov 23 '16 at 18:39