Finite orbits for nilpotent actions on the torus

TL;DR

This paper extends fixed point results to nilpotent groups of torus diffeomorphisms, showing they have finite orbits if some element has a non-zero Lefschetz number, generalizing classical fixed point theorems.

Contribution

It introduces a new fixed point theorem for nilpotent groups of torus diffeomorphisms based on Lefschetz number conditions, broadening classical results.

Findings

Nilpotent group actions on the 2-torus have finite orbits under certain Lefschetz conditions.

Extension of fixed point theorems from individual homeomorphisms to group actions.

Identification of conditions ensuring finite orbits for nilpotent group actions.

Abstract

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.