Journal of Lie Theory

\def\g{{\frak g}} Suppose that $G$ is a compact, connected, simple, exceptional Lie group with Lie algebra $\g$. We determine the sharp minimal exponent $k_{0}$, which depends on $G$ or $\g$, such that the convolution of any $k_{0}$ continuous, $G$-invariant measures is absolutely continuous with respect to Haar measure. The exponent $k_{0}$ is also the minimal integer such that any $k_{0}$-fold product of conjugacy classes in $G$ or $k_{0}$-fold sum of adjoint orbits in $\g$ has non-empty interior. Unlike in the classical case, the answer can be less than the rank of $G$ or $\g$.\par We also establish a dichotomy for orbital measures $\mu$, supported on non-trivial conjugacy classes or adjoint orbits of minimal non-zero dimension: for each $k$, either $\mu^{k}\in L^{2}$ or $\mu^{k}$ is singular with respect to Haar measure.