Subquadratic growth and uniform property \(\Gamma\)

Abstract

We prove that every unital separable ASH algebra with subquadratic growth has uniform property $\Gamma$ whenever it has no nonzero finite-dimensional representations. When simple and non-elementary, these algebras therefore satisfy the Toms--Winter regularity conjecture despite the fact that they generally fail its three conjecturally equivalent properties. In light of the second author's recent construction of a unital simple separable AH algebra of quadratic growth which fails uniform property $\Gamma$, we conclude that the quadratic dimension growth scale (equivalently, the 2-norm slow dimension growth scale) is the precise geometric threshold governing the potential failure of uniform property \(\Gamma\). We also extend recent work of Elliott--Niu and Vaccaro to the optimal subquadratic scale by proving that separable unital \(C^*\)-algebras with locally tracially subquadratic RSH approximation have uniform property \(\Gamma\), provided that they have no nonzero finite-dimensional representations.