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?
> 
> 
> 



Transpositions of order three
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](http://en.wikipedia.org/wiki/Complex_reflection_group#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.


