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May 24, 2021 at 10:39 comment added HJRW And indeed, every finite group arises as the isometry group (i.e. mapping class group) of a closed hyperbolic 3-manifold, by a 1988 theorem of Kojima.
May 22, 2021 at 20:12 comment added Student This is the first viewpoint - thanks! Do people know about how crazy it is? And this is also why I asked the second viewpoint - conceptually, unlike the first viewpoint which has many kinds of building blocks and even more kinds of combination (thus the craziness), the second viewpoint "only" uses links. Can one start with a link presentation $P$ for a link $L$ (with the $3$-fold obtained from surgery along $L$ denoted $M_L$), and build a group presentation from $P$ for the group $MCG(M_L)$? By the way, any reference/pointer will be appreciated :)
May 22, 2021 at 7:37 history answered Sam Nead CC BY-SA 4.0