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Let $n \ge 2$, then the Burnside group $B(n,m) = \langle a_1, \dots , a_n \mid \forall \omega, \ \omega^m = e \rangle$ is known to be non-amenable for odd $m \ge 665$, and finite for $m = 2,3,4,6$, but its finiteness for $m=5$ is open.

Question: Is the Burnside group $B(2,5)$ amenable?

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    $\begingroup$ Most likely this is unknown. $\endgroup$ – Moishe Kohan Sep 1 '16 at 3:26

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