Balance and Abelian complexity of the Tribonacci word

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism τ:0↦01, 1↦02 and 2↦0 is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in R2, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity ρ(n) of the Tribonacci word t which is the unique fixed point of τ. We show that ρ(n)∈{3,4,5,6,7} for each n⩾1. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all factors U and V of t of equal length, and for every letter a∈{0,1,2}, the number of occurrences of a in U and the number of occurrences of a in V differ by at most 2. While this result is announced in several papers, to the best of our knowledge no proof of this fact has ever been published. We offer two very different proofs: The first uses the word combinatorial properties of the generating morphism, while the second exploits the spectral properties of the incidence matrix of τ. Although we show that ρ(n) assumes each value 3⩽i⩽7, the sequence (ρ(n))n⩾1 itself seems to be rather mysterious and may reflect some deeper properties of the Rauzy fractal.