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The Nielsen realization problem. Let $S$ be a compact oriented topological surface and let $\text{Mod}(S)$ be its mapping class group, ie the group of orientation preserving diffeomorphisms of $S$ modulo isotopy. There is a natural surjection $\text{Diff}^+(S) \rightarrow \text{Mod}(S)$. The Nielsen realization problem was the conjecture (due to Jacob Nielsen) that every finite subgroup of $\text{Mod}(S)$ can be lifted to a finite subgroup of $\text{Diff}^+(S)$ (and thus is a subgroup of the group of automorphisms of a Riemann surface).

Nielsen proved this for finite cyclic subgroups (this is very nontrivial!), and a number of other people slowly chipped away at other classes of finite subgroups. In 1959, Kravetz published a paper which purported to prove that Teichmuller space is negatively curved. A "center of mass" argument would then establish that every finite subgroup of $\text{Mod}(S)$ fixes a point in Teichmuller space, and it then follows easily that the finite subgroup can be lifted to $\text{Diff}^+(S)$.

This was an important result, and Kravetz's paper was frequently quoted. However, in 1971 Linch pointed out in his thesis that Kravetz's paper had an error! In fact, in his 1974 thesis Howie Masur proved that Teichmuller space is not negatively curved (in a pretty strong sense).

Finally, in 1980 Steve Kerckhoff proved that Teichmuller space, while not negatively curved, did satisfy a subtle negative-curvature like property which gave the desired result.

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The Nielsen realization problem. Let $S$ be a compact oriented topological surface and let $\text{Mod}(S)$ be its mapping class group, ie the group of orientation preserving diffeomorphisms of $S$ modulo isotopy. There is a natural surjection $\text{Diff}^+(S) \rightarrow \text{Mod}(S)$. The Nielsen realization problem was the conjecture (due to Jacob Nielsen) that every finite subgroup of $\text{Mod}(S)$ can lifted to a finite subgroup of $\text{Diff}^+(S)$ (and thus is a subgroup of the group of automorphisms of a Riemann surface).

Nielsen proved this for finite cyclic subgroups (this is very nontrivial!), and a number of other people slowly chipped away at other classes of finite subgroups. In 1959, Kravetz published a paper which purported to prove that Teichmuller space is negatively curved. A "center of mass" argument would then establish that every finite subgroup of $\text{Mod}(S)$ fixes a point in Teichmuller space, and it then follows easily that the finite subgroup can be lifted to $\text{Diff}^+(S)$.

This was an important result, and Kravetz's paper was frequently quoted. However, in 1971 Linch pointed out in his thesis that Kravetz's paper had an error! In fact, in his 1974 thesis Howie Masur proved that Teichmuller space is not negatively curved (in a pretty strong sense).

Finally, in 1980 Steve Kerckhoff proved that Teichmuller space, while not negatively curved, did satisfy a subtle negative-curvature like property which gave the desired result.