On Area Growth in Sol

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 $e^{-2z} dx^2 + e^{2z} dy^2 + dz^2.$ Building on previous joint work with Matei Coiculescu, which characterizes the cut locus in Sol, we prove that the sphere of radius r in sol has area at most $20 πe^r$ provided that r is sufficiently large. This estimate is sharp up to a factor of 10

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