I was trying to read an old paper concerning equations on free groups and immediately came to a puzzling statement which made me wonder whether I am fundamentally misunderstanding something or if I am unaware of some notational convention.
The "trivial example" that if $w(x)=xa_1 a^{-1} a_1^{-1}$, the solutions $u$ are all elements $a_1^v$ where $v$ is an integer does not make sense to me. Firstly, what is $a$ as it was never formally defined? Apparently we are working with the free group on $\{x,a_1,\ldots, a_r\}$ and considering solutions in the free subgroup generated by $\{a_1,\ldots, a_r\}$. So, is $a$ just an arbitrary element in that free subgroup? If so, I do not see how the equation could hold for $x=a_1^v$. It seems to me that $w(x)=1$ and $x=a_1^v$ requires $a$ to be $a_1^{-v}$. But, I thought that the only variable is $x$, with the other letters representing fixed group elements.
What am i missing?