Braid Equivalence (awaiting review)

Given a group $G$, the braid operations on an $n$-tuples of elements in $G$ are the maps

$$Q_i: (g_1, \ldots, g_{i-1}, g_i, g_{i+1}, \ldots, g_n) \to (g_1, \ldots g_{i-1}, g_i+1, g_{i+1}^{-1}g_ig_{i+1}, \ldots, g_n)$$

and their inverses, for $1 \leq i \leq n-1$. These operations generate an action on generating vectors of one refined passport.

Two generating vectors from the same refined passport are considered braid equivalent if they are in the same orbit under this action.