Fixed points of nilpotent actions on surfaces of negative Euler characteristic

TL;DR

This paper proves that locally nilpotent groups of $C^{1}$ diffeomorphisms on compact surfaces with negative Euler characteristic have finite orbits bounded by the surface's Euler characteristic, with some orbits consisting of fixed points.

Contribution

It establishes bounds on the size of finite orbits for nilpotent diffeomorphism groups on surfaces with negative Euler characteristic and identifies conditions for fixed points.

Findings

Finite orbits are bounded by the Euler characteristic.

Existence of global contractible fixed points for certain subgroups.

Boundedness depends on the surface's topology.

Abstract

We prove that a locally nilpotent group $G$ of $C^{1}$ diffeomorphisms of a compact surface $S$ of non-vanishing Euler characteristic has a finite orbit ${\mathcal O}$ whose cardinal is bounded by above by a function of the characteristic of Euler of $S$. We focus on the case of negative Euler characteristic $\chi (S)$. Then we can choose ${\mathcal O}$ so that it consists of global contractible fixed points of thesubgroup $G_0$ of $G$ consisting of isotopic to the identity elements. In particular $G$ has a global contractible fixed point if it consists of isotopic to the identity elements.