Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound

Abstract: We establish a compactness theorem for the resolvent of free Laplacians. More precisely, let $X=(X_1,\ldots,X_n)$ be a self-adjoint tuple admitting Lipschitz conjugate variables in the sense of Dabrowski (2014). Under the non-commutative curvature--dimension condition $CD(c,\infty)$ with $c&gt;0$, introduced in our previous work (2026), we prove that the free Laplacian $Δ:=\partial<sup>*\bar\partial$ is proper, that is, has compact resolvent. Our argument is non-perturbative and uses a spectral variant of the free Bochner method based on Dabrowski's almost-commutation formulas. A further stochastic argument gives a normal unital trace-preserving embedding of $M=W<sup>*(X_1,\ldots,X_n)$ into $L(\mathbb F_2)$. Together, the compactness and embedding theorems gives several operator-algebraic consequences, including the Haagerup approximation property, the CMAP, and strong solidity. We also obtain proper proximality and bi-exactness. We finally obtain several geometric consequences. We prove a free analogue of Klartag's improved log--Lichnerowicz inequality (2023), together with rigidity in the equality case. More surprisingly, under $CD(0,\infty)$, we prove a dimension-free Kannan--Lovász--Simonovits (KLS) bound in the free setting. As a consequence, we obtain a free thin-shell estimate for isotropic tuples. We conclude with further questions on spectral comparison with the canonical semicircular model.