Fixed point properties of nilpotent and solvable Lie group actions on Hadamard manifolds

TL;DR

This paper proves fixed point properties for actions of nilpotent and solvable Lie groups on nonpositively curved compact manifolds, extending understanding of group actions in geometric contexts.

Contribution

It establishes new fixed point results for nilpotent and solvable Lie group actions on Hadamard manifolds, highlighting conditions for fixed points and periodic points.

Findings

Nilpotent Lie group actions have fixed points on nonpositively curved compact manifolds.

Solvable Lie group actions have either fixed points or 2-periodic points.

Results extend fixed point theory in geometric group actions.

Abstract

We will prove the following theorems. The first theorem posits the existence of a fixed point for the actions of nilpotent Lie groups on nonpositively curved compact manifolds. The second theorem states that actions of solvable Lie groups on nonpositively curved compact manifolds have either a fixed point or a 2-periodic point.