On the 2-abelian complexity of the Thue–Morse word

We show that the 2-abelian complexity of the infinite Thue–Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue–Morse word.