## local fundamental group of elliptic singularities

Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ is Gorenstein ($K_X$ is Cartier) and $R^1\pi_* \mathcal{O}_Y=\mathbb{C}_x$ where $\pi:Y\to X$ is a resolution of the singularity.

Equivalently: $\pi_*\omega_Y=m_x\omega_X$.

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 To motivate my question, recall that the rational Gorenstein singularities of surfaces are exactly the ADE singularities and in this case the local fundamental group is finite. – Benoît Nov 3 at 9:00