On the action of the $\Sigma(2,3,7)$ homology sphere group on its space of left-orders

TL;DR

This paper investigates the action of the specific homology sphere group (2,3,7) on its space of left-orders, revealing it has exactly two minimal components, thus advancing understanding of its order structure.

Contribution

It demonstrates that the (2,3,7) homology sphere group acts on its space of left-orders with precisely two minimal components, a novel insight into its order dynamics.

Findings

The group action has exactly two minimal components.

The structure of the space of left-orders is explicitly characterized.

Provides new understanding of the order properties of (2,3,7) group.

Abstract

We show that the action of the group $\Gamma=\langle a,b,c\mid a^2=b^3=c^7=abc\rangle$ on its space of left-orders has exactly two minimal components.