Let Sol be the $3$--dimensional solvable Lie group whose underlying space is $mathbb R^3$ and whose left-invariant Riemannian metric is given by e^{-2z} ,dx^2 + e^{2z} ,dy^2 + dz^2. Let $Eco mathbb R^3 to operatorname{Sol} $ be the Riemannian exponential map. Given $V=(x,y,z) in mathbb R^3$, let $gamma_V={E(tV)mid t in 0,1}$ be the corresponding geodesic segment. Let AGM stand for the arithmetic--geometric mean. We prove that $gamma_V$ is a distance-minimizing segment in Sol if and only if operatorname{AGM} bigl(sqrt{xy},tfrac{1}{2}sqrt{(x+y)^2+z^2}bigr) leq pi. We use this inequality to precisely characterize the cut locus in Sol, prove that the metric spheres in Sol are topological spheres, and almost exactly characterize their singular sets.
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