  Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that
at every position of a Sturmian word starts an abelian power of exponent $k$
for every $k > 0$. We improve on this result by studying the maximum exponents
of abelian powers and abelian repetitions (an abelian repetition is an analogue
of a fractional power) in Sturmian words. We give a formula for computing the
maximum exponent of an abelian power of abelian period $m$ starting at a given
position in any Sturmian word of rotation angle $\alpha$. vAs an analogue of
the critical exponent, we introduce the abelian critical exponent $A(s_\alpha)$
of a Sturmian word $s_\alpha$ of angle $\alpha$ as the quantity $A(s_\alpha) =
limsup\ k_{m}/m=limsup\ k'_{m}/m$, where $k_{m}$ (resp. $k'_{m}$) denotes the
maximum exponent of an abelian power (resp.~of an abelian repetition) of
abelian period $m$ (the superior limits coincide for Sturmian words). We show
that $A(s_\alpha)$ equals the Lagrange constant of the number $\alpha$. This
yields a formula for computing $A(s_\alpha)$ in terms of the partial quotients
of the continued fraction expansion of $\alpha$. Using this formula, we prove
that $A(s_\alpha) \geq \sqrt{5}$ and that the equality holds for the Fibonacci
word. We further prove that $A(s_\alpha)$ is finite if and only if $\alpha$ has
bounded partial quotients, that is, if and only if $s_{\alpha}$ is
$\beta$-power-free for some real number $\beta$. Concerning the infinite
Fibonacci word, we prove that: i) The longest prefix that is an abelian
repetition of period $F_j$, $j>1$, has length $F_j( F_{j+1}+F_{j-1} +1)-2$ if
$j$ is even or $F_j( F_{j+1}+F_{j-1} )-2$ if $j$ is odd, where $F_{j}$ is the
$j$th Fibonacci number; ii) The minimum abelian period of any factor is a
Fibonacci number. Further, we derive a formula for the minimum abelian periods
of the finite Fibonacci words
