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Jan 3, 2020 at 3:16 comment added YCor @BenjaminSteinberg (You mean test finite order.) Indeed, you're right. It looks like a simpler approach.
Jan 3, 2020 at 0:49 comment added Benjamin Steinberg Can't you test finiteness in a hyperbolic group without using the conjugacy problem? Since there are only finitely many finite conjugacy classes of finite order there is a fixed number $n$ so that every element of finite order has order dividing $n$. To check finiteness you just raise an element to the power $n$ and use the word problem to check if you get $1$.
Jan 2, 2020 at 3:35 comment added Misha Ah, I see. Thank you!
Jan 1, 2020 at 23:10 comment added YCor (Please don't accept this answer, because the freeness problem in hyperbolic groups is more interesting.)
Jan 1, 2020 at 23:09 history answered YCor CC BY-SA 4.0