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 generaed by letters $\sigma_1,\dots,\sigma_{n-1}$ subject to relations $$ \begin{aligned} &\sigma_i\sigma_j=\sigma_j\sigma_i, && 1\leq i,j<n, |i-j|>1;\\\\ &\sigma_i\sigma_j\sigma_i=\sigma_j\sigma_i\sigma_j, &&1\leq i,j<n, |i-j|=1; \\\\ &\sigma_i^2=1, && 1\leq i<n \end{aligned} $$ 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 $$ \begin{aligned} &\sigma_i^3=1, && 1\leq i<n \end{aligned} $$ 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?