I am trying to understand the structure of conjugacy classes in the Grigorchuk group because it is related to the derivations in group algebra (for example, see this). I am particularly interested in whether there is an infinite conjugacy class in this group, meaning if the Grigorchuk group is an FC group. I believe that it is not, but I have not found any statements about this.
I also wonder if there is an explicit way to write a word $w$ such that $l(w) \geq n$, where $l(w)$ is the length of the word. That is, if there is a way to generate long words? Another question is there an algorithm that allows us to calculate the length of a word? In Harpe's book, I saw that the word problem is solvable in this group, which implies that we can check if any shorter word is equal to a given word. But maybe there are more convenient ways to calculate word's length?
Any help/thoughts are appreciated. I would also be grateful if someone can provide me with literature related to these questions.