**Title:** Discrete subgroups of Diff(I) and the extension of Hölder's Theorem

**Speaker:** Azer Akhmedov

**Speaker Info:** North Dakota State University

**Brief Description:**

**Special Note**:

**Abstract:**

It is a classical result (essentially due to O.Hölder, from the year 1901) that if G is a subgroup of Homeo(R) such that every nontrivial element acts freely then G 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 (Solodov, Barbot, Kovacevic, Farb-Franks) that in this case the group is metabelian, in fact, it is isomorphic to a subgroup of the affine group Aff(R).We answer this question for an arbitrary N assuming some regularity on the action of the group.

The following theorem is from our published paper in 2016.

Theorem 1. Let G be a subgroup of Diff^{1+\epsilon }(I) [of Diff^{2}(I)] such that every nontrivial element of G has at most N fixed points. Then G is solvable [metabelian].

By sharpening the methods, we succeed proving the same and even a stronger claim for subgroups of Diff(I).

Theorem 2. Let G be a subgroup of Diff(I) such that every nontrivial element of G has at most N fixed points. Then G is isomorphic to a subgroup of the affine group Aff(R).

The proof is based on our study of C_0-discrete subgroups of Diff(I) from an earlier work. In the talk, we will briefly mention a few other results that we have obtained as a product of our study of discrete subgroups of Diff(I).

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