A bound for diameter of arithmetic hyperbolic orbifolds

Let $O$ be a closed $n$-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of $O$ is bounded above by $\frac{c_1\log vol(O) + c_2}{h(O)},$ where $h(O)$ is the Cheeger constant of $O$, $vol(O)$ is its volume, and constants $c_1$, $c_2$ depend only on $n$.

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