Uniform Comparison of Hyperbolic Ball Volumes on the Universal Cover

Let $\mathrm{\parallel }M{\mathrm{\parallel }}_{\mathrm{\Delta }}$ denote the simplicial volume of $M$, $V_r(X,h)={\sup }_{x\in X}{\mathrm{Vol}}_h(B_h(x,r))$, and ${\mathbb{H}}^n$ denotes hyperbolic $n$-space. We prove that, if a closed oriented $n$-manifold $M$ admits a hyperbolic metric, then there is a dimensional constant ${\delta }_n>0$ such that every Riemannian metric $g$ on $M$ with $$\frac{{\mathrm{Vol}}_g(M)}{\mathrm{\parallel }M{\mathrm{\parallel }}_{\mathrm{\Delta }}}<{\delta }_n$$ satisfies $$V_r(\tilde{M},\tilde{g})\ge V_r({\mathbb{H}}^n)\text{for every }r\ge 1.$$