Abstract
G.~Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau:$ $0\mapsto 01,$ $1\mapsto 02$ and $2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $\reals^2,$ each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity $\abp(n)$ of the Tribonacci word $\tribo$ which is the unique fixed point of $\tau.$ We show that $\abp(n)\in\{3,4,5,6,7\}$ for each $n\geq 1.$ Our proof relies on the fact that the Tribonacci word is $2$-balanced, i.e., for all factors $U$ and $V$ of $\tribo$ of equal length, and for every letter $a\in \{0,1,2\},$ 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 $\tau.$ Although we show that $\abp(n)$ assumes each value $3\leq i\leq 7,$ the sequence $(\abp(n))_{n\geq 1}$ itself seems to be rather mysterious and may reflect some deeper properties of the Rauzy fractal.