Fixed points of local actions of nilpotent Lie groups on surfaces

TL;DR

This paper proves that for nilpotent Lie group actions on surfaces, the presence of a nonzero fixed-point index in a neighborhood guarantees a fixed point of the entire group within that neighborhood.

Contribution

It establishes a new fixed point existence result for local actions of nilpotent Lie groups on surfaces based on fixed-point index conditions.

Findings

Nonzero fixed-point index implies existence of a fixed point of the group.

The result applies to local actions of connected nilpotent Lie groups on surfaces.

Provides conditions under which fixed points must exist for group actions.

Abstract

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of $\varphi_t|U$ is nonzero for sufficiently small $t>0$, then ${\rm Fix} (G) \cap K \ne \emptyset$.