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

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 +(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. Our main result is the following theorem. Theorem 0.1.(Main Theorem) Let ∈ (0, 1) and Γ be a subgroup of Diff 1+ + (I) such that every nontrivial element of Γ has at most N fixed points. Then Γ is solvable. Assuming a higher regularity on the action we obtain a stronger result Theorem 0.2. Let Γ be a subgroup of Diff +(I) such that every nontrivial element of Γ has at most N fixed points. Then Γ is metaabelian. An important tool in obtaining these results is provided by Theorems B-C from [A]. Theorem B (Theorem C) states that a non-solvable (nonmetaabelian) subgroup of Diff 1+ + (I) (of Diff 2 +(I)) is non-discrete in the C0 metric. Existence of C0-small elements in a group provides effective tools in tackling the problem. Such tools are absent for less regular actions, and for the group Homeo+(I) (even for Diff+(I)), the problem of characterizing subgroups where every non-identity element has at most N ≥ 2 fixed points still remains open. Basic Notations: Throughout this paper, G will denote the group Diff 1+ + (I) where ∈ (0, 1). We let Γ ≤ G. For every g ∈ Γ, Fix(g) will 1