On groups of diffeomorphisms of the interval with finitely many fixed points

TL;DR

This paper proves that groups of orientation-preserving diffeomorphisms of the interval with finitely many fixed points are either affine or have infinite girth, extending the Girth Alternative to a broader class of groups.

Contribution

It establishes the Girth Alternative for a larger class of diffeomorphism groups, showing they are either affine or have infinite girth, using advanced tools like local transitivity.

Findings

Groups are either affine or have infinite girth.

Local transitivity is proved for non-affine groups with finitely many fixed points.

The Girth Alternative holds for these groups.

Abstract

In [13], it is proved that any subgroup of $\mathrm{Diff}_{+}^{\omega }(I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether or not the Girth Alternative holds for subgroups of $\mathrm{Diff}_{+}^{\omega }(I)$. In this paper, we answer this question affirmatively for even a larger class of groups of orientation preserving diffeomorphisms of the interval where every non-identity element has finitely many fixed points. We show that every such (irreducible) group is either affine (in particular, metaabelian) or has infinite girth. The proof is based on our study of discrete subgroups of the diffeomorphism group $\mathrm{Diff}_{+}(I)$ which we initiated in [9] and later developed in [1] and [2]; more specifically, our results are obtained by sharpening the tools from the earlier works [1] and [2]. One of the major tools (local transitivity) is heavily exploited in [2] to get an extension of Holders theorem which is crucially used in this paper. We show that local transitivity can be proved for any (up to a conjugacy) non-affine group of irreducible diffeomorphisms with every non-identity element having finitely many fixed points.