Surface groups of diffeomorphisms of the interval

TL;DR

This paper demonstrates the existence of surface groups within the diffeomorphism group of the interval, exhibiting complex dynamical properties such as topological transitivity and minimal stabilizers.

Contribution

It constructs specific surface groups acting on the interval with detailed dynamical features, advancing understanding of group actions on one-dimensional manifolds.

Findings

Existence of surface groups in Diff([0,1]) with no global fixed point

Actions are topologically transitive on (0,1)

Only countably many points have non-trivial stabilizer

Abstract

We prove that the group of diffeomorphisms of the interval $[0,1]$ contains surface groups whose action on $(0,1)$ has no global fix point, is topologically transitive, and such that only countably many points of the interval $(0,1)$ have non-trivial stabiliser.