Heng Zhang

Uniform Comparison of Hyperbolic Ball Volumes on the Universal Cover

Let $\|M\|_{\Delta}$ denote the simplicial volume of $M$, $V_r(X,h)=\sup_{x\in X}\operatorname{Vol}_h\big(B_h(x,r)\big)$, 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 Ri...