Let \({\mathcal {O}}\) be a closed n-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of \({\mathcal {O}}\) is bounded above by

$$\begin{aligned} \frac{c_1\log \mathrm {vol}({\mathcal {O}}) + c_2}{h({\mathcal {O}})}, \end{aligned}$$

where \(h({\mathcal {O}})\) is the Cheeger constant of \({\mathcal {O}}\), \(\mathrm {vol}({\mathcal {O}})\) is its volume, and constants \(c_1\), \(c_2\) depend only on n.

中文翻译：

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算术双曲单径的直径的界

令\（{\ mathcal {O}} \）是一个封闭的n维算术（实数或复数）双曲单值。我们证明\（{\ mathcal {O}} \）的直径由

$$ \ begin {aligned} \ frac {c_1 \ log \ mathrm {vol}（{\ mathcal {O}}）+ c_2} {h（{\ mathcal {O}}）}}，\ end {aligned} $$

其中\（h（{\ mathcal {O}}）\）是\（{\ mathcal {O}} \），\（\ mathrm {vol}（{\ mathcal {O}}）\）的Cheeger常数是其体积，常数\（c_1 \），\（c_2 \）仅取决于n。