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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asked Apr 14, 2017 at 14:38
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7$\begingroup$ In the arithmetic case, the smallest orbifold has Euler characteristic of the order of $10^{-15}$ and dimension 16, see numdam.org/article/ASNSP_2004_5_3_4_749_0.pdf $\endgroup$– Bruno MartelliApr 14, 2017 at 16:50
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11$\begingroup$ For hyperbolic 2-orbifolds, this is connected to open questions in number theory. The euler characteristic of a triangle orbifold is -1+1/p+1/q+/1r. Then one can realize a rational $-1+4/n (n>4)$ iff the Erdös-Strauss conjecture holds. en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture $\endgroup$– Ian AgolApr 14, 2017 at 17:20
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