  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 $R^2$, each domain being translated by the same vector modulo a
lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci
word $t$ which is the unique fixed point of $\tau$. We show that $AC(n)\in
{3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed.
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 $a \in
{0,1,2}$, 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 of the 2-balance property of $t$.
The first uses the word combinatorial properties of the generating morphism,
while the second exploits the spectral properties of the incidence matrix of
$\tau$.
