A proper CAT(0) space whose 2-sphere filling inequality has constant below 1/(6*sqrt(pi)) satisfies isoperimetric inequalities with exponent 1+delta for every delta>0, equivalent to asymptotic rank at most 2.
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Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature
A proper CAT(0) space whose 2-sphere filling inequality has constant below 1/(6*sqrt(pi)) satisfies isoperimetric inequalities with exponent 1+delta for every delta>0, equivalent to asymptotic rank at most 2.