Classification of connected proper pairs in the affine transformation group

TL;DR

This paper classifies all connected proper pairs of subgroups within the affine transformation group of the plane, extending previous subgroup classifications to pairs acting properly on the plane.

Contribution

It provides a complete list of connected closed proper pairs in the affine group of 2, extending Kobayashi's classification of subgroups acting properly.

Findings

Complete classification of connected proper pairs in the affine group of 2.

Extension of Kobayashi's subgroup classification to pairs.

Identification of all pairs with proper action on 2.

Abstract

Let $(L, H)$ be closed subgroups of a locally compact group $G$. The pair $(L, H)$ is said to be proper if the action of $L$ on the homogeneous space $G/H$ is proper. We give a complete list of connected closed proper pairs in the affine transformation group of $\mathbb{R}^2$. This result extends Kobayashi's classification (1992) of connected closed subgroups of the affine transformation group of $\mathbb{R}^2$acting properly on $\mathbb{R}^2$.