I'm currently working with a group given by the presentation $\bigl
a) contains no inverse symbols and
b) does not contain T more than once in a row
also either ends in T or has a suffix equivalent to T.
My thought is that any string equivalent to the given string is made by multiplying by conjugates of $RTRTRT$ and $TT$, but in order for the resulting string not to contain any inverse symbols, if you multiply by $ARTRTRTA^{-1}$ for an element $A$ then the string you're multiplying by must start with $A$ so that the inverse cancels it out. But this will be the same as if you insert the $RTRTRT$ after the initial string $A$. Also, since one can't have two $T$'s in a row, inserting $TT$ is not allowed, and any pair of $T$'s must be canceled upon appearing in the string.
So now, if the only way equivalent strings can be made is by inserting $RTRTRT$ somewhere in the string and canceling $T$'s, it would appear that all of the strings equivalent to one ending with T also have a suffix equivalent to T.
Does this reasoning work? If not, how can I fix it?