What does "subgroup of $F_n$ consisting of all words of exponent sum $k$" mean?

Question

I am currently reading the following paper by Birman and Hilden, https://www.jstor.org/stable/1970830?casa_token=7y0vMR6x_lsAAAAA:AObJdZWkgbTOaDbo5woR_Zm01B2xYRNo1aW1j605DqwmFyZfT83cJIngRHDZtAmGR2YymSASh4xx19f8lEaZxlx1Ox2yIR6aS1grOkgTOB5Kc9euIEVc On page 436 after Lemma 7.1 the author says "Let $H$ be a subgroup of index $k$ in $F_n$ consisting of all words of exponent sum $k$ in $x_1,x_2\dots x_n$." What subgroup is being talked about? Is the "exponent sum" taken over the entire word or each generator individually? I think they mean sybgroup generated by words of exponent length $k$ but I have not been able to prove that it is a index $k$ subgroup. Any help would be appreciated.

Answer 1

It seems that it should say "exponent sum divisible by $k$". This is a subgroup of index $k$ because it is exactly the kernel of the surjective homomorphism $F_n\to\mathbb{Z}/(k)$ which sends each generator to $1$ (so it sends a word to the sum of its exponents mod $k$).