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 $\sigma_{i}^{6}$.

  • $\begingroup$ "normal subgroup generated by $x$" = "subgroup generated by conjugates of $x$" ("normal subgroup generated by conjugates of $x$" is redundant) $\endgroup$ – YCor Nov 22 '16 at 16:33
  • $\begingroup$ 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. $\endgroup$ – Misha Nov 22 '16 at 17:03
  • 3
    $\begingroup$ 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. $\endgroup$ – Derek Holt Nov 22 '16 at 19:14
  • $\begingroup$ Misha. Can you give a reference for your claim that it is known for certain values? $\endgroup$ – Joseph Van Name Nov 23 '16 at 0:01
  • $\begingroup$ 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. $\endgroup$ – Misha Nov 23 '16 at 18:39

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.