EXTENSION OF HÖLDER’S THEOREM IN Diff 1++ (I)

Abstract: We prove that if Γ is subgroup of Diff 1++ (I) and N is a natural number such that every non-identity element of Γ has at most N fixed points then Γ is solvable. If in addition Γ is a subgroup of Diff 2+(I) then we can claim that Γ is metaabelian. It is a classical result (essentially due to Hölder, cf.[N]) that if Γ is a subgroup of Homeo+(R) such that every nontrivial element acts freely then Γ is Abelian. A natural question to ask is what if every nontrivial element has at most N fixed points where N is a fixed natural number. In the case of N = 1, we do have a complete answer to this question: it has been proved by Solodov (not published), Barbot [B], and Kovacevic [K] that in this case the group is metaabelian, in fact, it is isomorphic to a subgroup of the affine group Aff(R). (see [FF] for the history of this result, where yet another nice proof is presented). In this paper, we answer this question for an arbitrary N assuming some regularity on the action of the group