I was reading a paper on the theory of braids,and it states
"Garside proved that two positive words in $B_n$ are equivalent iff they are equivalent in $B_n^+$".
I just want to know what equivalent words mean. Does it mean something specific in the braid group, or do we have a general definition for equivalent words in any group.?
I looked up Garside's paper, but he hasn't used the word equivalent anywhere.
On
They represent the same element. The famous word problem is to determine if two words represent the same element. My crude understanding is that there is no algorithm for doing this.
If a group $G$ is generated by $g_1,g_2,\ldots,g_n$, then two words over $g_i^{\pm}$ are equivalent in $G$ if and only if they define the same element in $G$. For example, if $g_1g_2$ has order $3$ in $G$, then the words $g_1g_2g_3$ and $g_1^{-1}g_2^{-1}g_3^{-1}$ are equivalent in $G$.
The braid group $B_n$ has $n-1$ standard generators $s_1,s_2,\ldots,s_{n-1}$. The positive words over $s_i$ (i.e. those that do not involve $s_i^{-1}$) form a submonoid $B_n^+$ of $B_n$.
Let $w_1$ and $w_2$ be two positive words over $s_i$. Then if $w_1$ and $w_2$ are equivalent in the monoid $B_n^+$, that is they represent the same element of $B_n^+$, then they are certainly equivalent in the group $B_n$. That is true in any submonoid of any group.
But the converse is not true in general for submonoids of groups. Garside's result is that it is true for braid groups. That is, if two positive words are equivalent in $B_n$, then they are also equivalent in $B_n^+$.