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.
Singular (Lipschitz) homology and homology of integral currents
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abstract
We compare the homology groups $H_n ^{IC}(X)$ of the chain complex of integral currents with compact support of a metric space $X$ with the singular Lipschitz homology $H^L_n (X)$ and with ordinary singular homology. If $X$ satisfies certain cone inequalities all these homology theories coincide. On the other hand, for the Hawaiian Earring the homology of integral currents differs from the singular Lipschitz homology and it differs also from the classical singular homology $H_n(X)$.
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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.