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For 3-manifolds, the leading coefficient of the $L^2$-Alexander torsion is anumerical invariant of a real first cohomology class. We show that the leadingcoefficient equals the relative $L^2$-torsion of the manifold cut up along anorm-minimizing surface dual to the cohomology class. Furthermore, the leadingcoefficient equals the relative $L^2$-torsion of the guts associated to thecohomology class. Finally, we prove that the leading coefficient is constant onany open Thurston cone. The main ingredients are a new criterion for theconvergence of Fuglede-Kadison determinants and the work of Agol and Zhang onguts of 3-manifolds.

Guts determine the leading coefficients of $L^2$-Alexander torsions