[Paper Review] Fixed points of discrete nilpotent group actions on S^2

S. Druck, Fuquan Fang+1 more|ArXiv.org|Sep 3, 2001

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper establishes that discrete nilpotent group actions on the 2-sphere $S^2$, generated by $C^1$ diffeomorphisms sufficiently close to the identity (within a neighborhood $\ ilde{\mathcal{V}}_k$ depending on nilpotency length $k$), must have a fixed point. Furthermore, if such an action has a finite nontrivial orbit, it must have at least two fixed points, extending Bonatti’s theorem for commuting diffeomorphisms to higher-step nilpotent groups.

ABSTRACT

We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 has non-empty fixed point set. Moreover, the G-action has at least two fixed points if the action has a finite non-trivial orbit.

Motivation & Objective

• To extend Bonatti’s fixed point theorem for commuting $C^1$ diffeomorphisms on $S^2$ to discrete nilpotent group actions.
• To establish the existence of fixed points for $k$-nilpotent subgroups of $\mathrm{Diff}^1(S^2)$ generated by elements in a $C^1$-neighborhood of the identity.
• To prove that finite nontrivial orbits in such actions force at least two fixed points.
• To generalize Plante’s result on connected nilpotent Lie group actions to discrete nilpotent group actions on $S^2$.

Proposed method

• Construct a decreasing nested sequence of $C^1$-open neighborhoods $\mathcal{V}_k$ of the identity in $\mathrm{Diff}^1(S^2)$, independent of the number of generators.
• Use induction on the nilpotency length $k$, reducing the problem to lower-length nilpotent subgroups via the lower central series.
• Apply the Main Lemma to show that if $Fix(G_{(1)}, f_1, \dots, f_n)$ is nonempty and $f_{n+1}$ acts on it, then $Fix(G_{(1)}, f_1, \dots, f_{n+1})$ is nonempty.
• Use $\omega$-recurrent points and character curves associated with diffeomorphisms to locate fixed points in disks enclosed by these curves.
• Leverage the $f$-invariance of fixed point sets and Corollary 4.4 to iteratively construct fixed points within nested disks.
• Use the finite intersection property of closed fixed point sets to extend results from finite to infinite generating sets.

Experimental results

Research questions

• RQ1 Does a $k$-nilpotent subgroup of $\mathrm{Diff}^1(S^2)$ generated by elements $C^1$-close to the identity have a fixed point?
• RQ2 If such an action has a finite nontrivial orbit, must it have at least two fixed points?
• RQ3 Can Bonatti’s result for commuting diffeomorphisms be generalized to higher-step nilpotent groups on $S^2$?
• RQ4 Does the existence of a single element with finitely many fixed points in the center of a nilpotent group action imply multiple fixed points for the full action?
• RQ5 Can Plante’s theorem on connected nilpotent Lie group actions be extended to discrete nilpotent group actions on $S^2$?

Key findings

• For each $k \geq 2$, there exists a $C^1$-neighborhood $\mathcal{V}_k$ of the identity such that any $k$-nilpotent subgroup $G \subset \mathrm{Diff}^1(S^2)$ generated by elements in $\mathcal{V}_k$ has a nonempty fixed point set.
• If the $G$-action has a finite nontrivial orbit, then the fixed point set contains at least two points.
• The fixed point set of a $k$-nilpotent group action on $S^2$ is nonempty whenever the generators lie in $\mathcal{V}_k$, regardless of the number of generators.
• The existence of a single element $h$ in the center of $G$ with $2 \leq \#\mathrm{Fix}(h) < \infty$ implies that the full $G$-action has at least two fixed points.
• The results extend verbatim to $\mathbb{RP}^2$ via the universal covering $S^2 \to \mathbb{RP}^2$, preserving the fixed point conclusions.
• The theorems are applied to foliation theory, showing that $C^1$-small perturbations of trivial fibrations over $S^2$ or $\mathbb{RP}^2$ must have compact leaves near fibers.