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This is a follow-up to this question. Which rational numbers arise as Euler characteristics of orbifold quotients of $\mathbb{H}^n?$ The answer is not even clear for $n=2.$ It is clear that the Euler characteristic is negative, and is not bigger than that of the $2, 3, 7$ triangle orbifold (so, no smaller than $1/84$ in absolute value), but what the range is otherwise is not quite clear. In higher dimensions, an analogous result is shown by Adeboye and Wei, so for each fixed (even) dimension there is a lower bound on the absolute value of the Euler characteristic, but once the dimension is unrestricted, things are less clear.

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