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Let ( M, g ) be any compact, connected, Riemannian manifold of dimension  n . We use a transport of measures and the barycentre to construct a map from ( M, g ) onto a Hyperbolic manifold (ℍ n/ Λ,  g 0 ) (Λ is a torsionless subgroup of  Isom (ℍ n,g 0 )), in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of ( M, g ) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff  ε -approximation between (ℍ n/ Λ,  g 0 ) and ( M, g ). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if  ɛ &lt; 1 64   n 2 min ( inj ( ℍ n / Λ , g 0 ) , 1 ) $\varepsilon  &lt; {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$ , then Vol ( M n , g ) ≥ ( 1 + 160 n ( n + 1 ) ɛ min ( inj ( H n / Λ , g 0 ) , 1 ) ) n 2 | deg   h | ⋅ Vol ( X n , g 0 ) . $$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon  \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).}  \cr  }$$

Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II