Global fixed points for nilpotent actions on the torus

TL;DR

This paper extends fixed point results for isotopic to identity maps on the 2-torus to nilpotent groups, showing such groups have global fixed points under certain conditions, generalizing Franks' theorem.

Contribution

It provides a new fixed point theorem for nilpotent subgroups of isotopic to identity diffeomorphisms on the 2-torus, broadening the scope of previous results.

Findings

Nilpotent groups of irrotational diffeomorphisms have global fixed points.

The derived subgroup of such a nilpotent group also has a fixed point.

The results generalize classical fixed point theorems to a broader group context.

Abstract

An isotopic to the identity map of the $2$-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the $2$-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the $2$-torus has a global fixed point.