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Classification of connected proper pairs in the affine transformation group

Shunsuke Miyauchi

TL;DR

The paper delivers a complete classification of connected closed proper pairs $L$ and $H$ in the affine transformation group of $\mathbb{R}^2$ by analyzing two principal ambient groups: $GL_2(\mathbb{R})\ltimes \mathbb{R}^2$ and $SL_2(\mathbb{R})\ltimes \mathbb{R}^2$. It extends Kobayashi's foundational work by providing a detailed, case-by-case determination of propriety through the (CI) condition and Cartan-projection arguments, culminating in explicit criteria and lists up to the coarse equivalence $\sim$. The results include a finite classification for the $SL_2(\mathbb{R})$-based case and a continuum of equivalence classes for the $GL_2(\mathbb{R})$-based case, with precise parameter inequalities governing when pairs are proper. The methodology combines algebraic classification (CKZ24) with asymptotic and geometric-properness tools, offering a robust framework for understanding discontinuous actions in low-dimensional affine transformation groups. Overall, the work broadens the scope of Kobayashi-type properness criteria to non-reductive and non-nilpotent ambient groups, providing concrete, usable criteria for researchers studying affine group actions and their discontinuities.

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 $\mathbb{R}^2$.

Classification of connected proper pairs in the affine transformation group

TL;DR

The paper delivers a complete classification of connected closed proper pairs and in the affine transformation group of by analyzing two principal ambient groups: and . It extends Kobayashi's foundational work by providing a detailed, case-by-case determination of propriety through the (CI) condition and Cartan-projection arguments, culminating in explicit criteria and lists up to the coarse equivalence . The results include a finite classification for the -based case and a continuum of equivalence classes for the -based case, with precise parameter inequalities governing when pairs are proper. The methodology combines algebraic classification (CKZ24) with asymptotic and geometric-properness tools, offering a robust framework for understanding discontinuous actions in low-dimensional affine transformation groups. Overall, the work broadens the scope of Kobayashi-type properness criteria to non-reductive and non-nilpotent ambient groups, providing concrete, usable criteria for researchers studying affine group actions and their discontinuities.

Abstract

Let be closed subgroups of a locally compact group . The pair is said to be proper if the action of on the homogeneous space is proper. We give a complete list of connected closed proper pairs in the affine transformation group of . This result extends Kobayashi's classification (1992) of connected closed subgroups of the affine transformation group of acting properly on .
Paper Structure (31 sections, 27 theorems, 120 equations, 2 figures, 20 tables)

This paper contains 31 sections, 27 theorems, 120 equations, 2 figures, 20 tables.

Table of Contents

  1. Introduction
  2. The case $G=SL_2(\mathbb{R})\ltimes \mathbb{R}^2$
  3. The case $G=GL_2(\mathbb{R})$
  4. Main result (the case $G=GL_2(\mathbb{R})\ltimes \mathbb{R}^2$)
  5. Organization of the paper
  6. Preliminaries
  7. Properness relation $\pitchfork$ and equivalence relation $\sim$
  8. The (CI) condition and properness
  9. The notations $\lesssim_+, \asymp_+,\lesssim_\times, \asymp_\times$
  10. Classification of proper pairs in $G=SL_2(\mathbb{R})\ltimes \mathbb{R}^2$
  11. Classification of the Lie subalgebras of $\mathfrak{sl}_2(\mathbb{R})\ltimes \mathbb{R}^2$
  12. Equivalent subgroups in the sense $\sim$
  13. Properness of the connected pairs
  14. Non-equivalence of the subgroups with respect to the relation $\sim$
  15. Classification of proper pairs in $G=GL_2(\mathbb{R})$
  16. ...and 16 more sections

Key Result

Theorem 1.2

Figures (2)

  • Figure 1: the Cartan projection of $Z, A({\gamma}), U,B({\gamma})$
  • Figure 2: Inclusions among $L(A(\alpha),i),L(B(\beta),j)\subset GL_2(\mathbb{R})\ltimes \mathbb{R}^2$

Theorems & Definitions (62)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.4
  • Definition 3.1
  • ...and 52 more