An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper and every cycle in $G$ contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of $G$ is called the acyclic chromatic index of $G$ and is denoted by $a'(G)$. Fiamčik and independently Alon, Sudakov, and Zaks conjectured that $a'(G) \leq Δ(G)+2$, where $Δ(G)$ denotes the maximum degree of $G$. The best known general bound is $a'(G)\leq 4(Δ(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lovász Local Lemma to show that if $G$ contains no copy of a given bipartite graph $H$, then $a'(G) \leq 3Δ(G)+o(Δ(G))$. Moreover, for every $\varepsilon>0$, there exists a constant $c$ such that if $g(G)\geq c$, then $a'(G)\leq(2+\varepsilon)Δ(G)+o(Δ(G))$, where $g(G)$ denotes the girth of $G$.

12 pages, 2 figures. This version uses the Local Cut Lemma instead of the Local Action Lemma