Some Remarks on the Topology of Hyperbolic Actions of $\mathbb{R}^n$ on $n$-manifolds

TL;DR

This paper explores the topology of totally hyperbolic $R^n$-actions on compact $n$-manifolds, analyzing fixed points, hyperbolic domains, and specific cases like the 2-sphere, $S^3$, and $RP^3$, with combinatorial insights.

Contribution

It provides new results on the structure of hyperbolic actions, including fixed point counts, hyperbolic domain numbers, and explicit constructions in 3D manifolds.

Findings

Analysis of fixed points and hyperbolic domains for $R$-actions

Combinatorial properties of hyperbolic actions on $S^2$

Construction of hyperbolic actions on $S^3$ and $RP^3$

Abstract

This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the $\mathbb{R}$-action generated by a fixed vector of $\mathbb{R}^n$, that provides some results on the number of hyperbolic domains and the number of fixed points of the action. We study with more details the case of the 2-sphere, in particular we investigate some combinatorial properties of the associated 4-valent graph embedded in $S^2$. We also construct hyperbolic actions in dimension 3, on the sphere $S^3$ and on the projective space $\mathbb{RP}^3$.