A solution of the word problem for braid groups via the complex reflection group $G_{12}$

Abstract

It is known that if there exists a Gr\"{o}bner-Shirshov basis for a group $G$, then we say that one of the decision problem, namely the word problem, is solvable for $G$ as well. Therefore, as the main target of this paper, we will present a (non-commutative) Gr\"{o}bner-Shirshov basis for the braid group associated with the congruence classes of complex reflection group $G_{12}$ which will give us  normal forms of the elements of $G_{12}$ and so will obtain a new algorithm to solve the word problem over it.