Allow me to take advantage of your collective scholarliness...

The symmetric group $\mathbb S_n$ can be presented, as we all know, as the group freely generated by letters $\sigma_1,\dots,\sigma_{n-1}$ subject to relations
$$
\begin{aligned}
&\sigma_i\sigma_j=\sigma_j\sigma_i, && 1\leq i,j&lt;n, |i-j|>1;\\\\
&\sigma_i\sigma_j\sigma_i=\sigma_j\sigma_i\sigma_j, &&1\leq i,j&lt;n, |i-j|=1; \\\\
&\sigma_i^2=1, && 1\leq i&lt;n
\end{aligned}
$$
If we drop the last group of relations, declaring that the $\sigma_i$'s are involutions, we get the braid group $\mathbb B_n$. Now suppose I add to $\mathbb B_n$ the relations
$$
\begin{aligned}
&\sigma_i^3=1, && 1\leq i&lt;n
\end{aligned}
$$
and call the resulting group $\mathbb T_n$.


  
  This very natural group has probably shown up in the literature. Can you provide references to such appearances?
  In particular, is $\mathbb T_n$ finite?
  


Following up what was mentioned in the comments for $n$ up to $5$. In "Factor groups of the braid group" Coxeter showed that the quotient of the Braid group by the normal closure of the subgroup generated by $\{\sigma_i^k \ | \ 1\le i\le n-1\}$ is finite if and only if $$\frac{1}{n}+\frac{1}{k}>\frac{1}{2}$$
In your case ($k=3$) this translates to this group being infinite for $n\geq 6$.

P.S. For the same question on Artin braid groups one can use the classification of finite complex reflection groups. See for example the first reference there, "On complex reflection groups and their associated braid groups" by Broué, Malle and Rouquier.
