Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

Let $\mathrm{\Sigma }$ be a closed minimal surface immersed in a Riemannian 3-manifold carrying an orthonormal Killing frame. This class of ambient spaces includes Lie groups with a bi-invariant metric. In this paper, we prove that the sum of the Morse index and the nullity of $\mathrm{\Sigma }$ is bounded from below by a constant times its genus.