Connected Closed Proper Pairs in Aff(R²)

• Connected closed proper pairs are defined as two connected closed subgroups of a locally compact group where one subgroup acts properly on the homogeneous space of the other.
• The classification in Aff(R²) employs reduction to linear parts, Cartan projection criteria, and detailed case analysis using representation theory and Lie theoretic methods.
• Results extend earlier work by addressing non-translational cases and establishing explicit properness conditions via subgroup parameter inequalities.

A connected closed proper pair consists of two connected closed subgroups $(L, H)$ of a locally compact group $G$ such that $L$ acts properly on the homogeneous space $G/H$. In the context of affine transformation groups—specifically $$G = \operatorname{Aff}(\mathbb{R}^2) = \operatorname{GL}_2(\mathbb{R}) \ltimes \mathbb{R}^2$$—the central problem is to classify such pairs up to a natural equivalence, extending prior work that treated only the case when $H$ is the pure translation group. This classification interlinks deep structural properties of matrix groups, action properness, and equivalence under conjugation, relying on criteria from representation theory and Lie theory (Miyauchi, 17 Jan 2026).

1. Fundamental Notions: Properness and Equivalence

Let $G$ be locally compact, and $L, H \subset G$ closed subgroups. The pair $(L, H)$ is a proper pair—denoted $L \bowtie H$—if for every compact $S \subset G$ the intersection $L \cap S H S^{-1}$ is relatively compact in $G$. If $L \bowtie H$, then the left action of $L$ on $G/H$ is proper in the usual sense.

Two subgroups $L, L'$ are equivalent (written $L \sim L'$) if there is a compact set $S$ such that $L \subset S L' S^{-1}$ and $L' \subset S L S^{-1}$. This equivalence preserves properness: if $L \bowtie H$ and $L \sim L'$, then $L' \bowtie H$ as well.

A related but distinct notion is the (CI)-condition: $(L, H)$ satisfies (CI) if $L \cap g H g^{-1}$ is compact for all $g \in G$. Properness always implies (CI), although the converse fails in general.

2. Classification Strategy and Key Methods

The classification of connected closed proper pairs $(L, H)$ in $\operatorname{Aff}(\mathbb{R}^2)$ is performed modulo the equivalence $\sim$, focusing on the case where both subgroups are non-compact and connected.

• The reduction to linear parts: The structure of $L$ and $H$ is analyzed via their projections to $\operatorname{GL}_2(\mathbb{R})$, denoted $L(L)$ and $L(H)$ respectively.
• Kobayashi’s properness criterion (Cartan projection criterion): For reductive subgroups, $L \bowtie H$ in $\operatorname{GL}_2(\mathbb{R})$ if and only if their Cartan projections $\mu(L) \bowtie \mu(H)$ in the positive Weyl chamber $a^+$. This provides necessary and sufficient conditions in the reductive case.
• Subgroups with linear part in $$\operatorname{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2$$ are reduced to the $\operatorname{SL}_2$ case (Theorem 7.3, “sld-ed”). When one factor is not reducible, properness depends on both the (CI)-condition and restrictions on the translational part (RCI-criterion, Theorem 7.4).
• The classification requires a detailed case analysis based on the conjugacy classes of connected subalgebras of $\mathfrak{gl}_2 \ltimes \mathbb{R}^2$, using foundational work by Chapovskyi–Koval–Zhur.

3. Classification of Connected Closed Subgroups up to Equivalence

All non-compact, connected closed subgroups of $\operatorname{Aff}(\mathbb{R}^2)$, up to $\sim$, can be categorized as follows:

• Linear-only subgroups (no translations):
  • $Z = \{\operatorname{diag}(e^t, e^t)\}$
  • $$A(\alpha) = \{\operatorname{diag}(e^{(\alpha+1)t}, e^{(\alpha-1)t})\}$$, $\alpha \in \mathbb{R}$
  • $$U = \{ \left(\begin{smallmatrix} e^t & 0\ te^t & e^t\end{smallmatrix}\right) \}$$
  • $D = \{\operatorname{diag}(e^s, e^t)\}$
  • $$B' = \{ \left(\begin{smallmatrix} e^t&0\ s e^t & e^t \end{smallmatrix}\right) \}$$
• Reductive subgroups with translations:
  • $B(\alpha)$ generated by $A(\alpha)$ and a translation along $e_2$
  • $L(D,1)$ and $L(B',1)$: minimal extensions of $D$ and $B'$ by translation
• Subgroups within $$\operatorname{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2$$:
  • $S \cong A \ltimes \mathbb{R}$ (split torus with one translation)
  • $L \cong$ unipotent axis $\ltimes \mathbb{R}$
  • $M$, $N$, and $R^2$ (pure translations)

4. Main Classification Theorem and Explicit List of Proper Pairs

The exhaustive list of all pairs $(L, H)$ of non-compact connected closed subgroups of $\operatorname{Aff}(\mathbb{R}^{2})$ satisfying the properness condition $L \bowtie H$ falls into four main regimes:

Case	Linear Parts $L(L), L(H)$	Properness Condition
Both reductive in $\operatorname{GL}_2(\mathbb{R})$	$Z$, $U$, $A(\alpha)$, $B(\alpha)$	Table 1 below specifies, e.g. $A(\alpha) \bowtie A(\beta)$ iff $|\alpha| \ne |\beta|$; $B(\alpha) \bowtie B(\beta)$ iff $|\alpha| < |\beta|$, etc.
Mixed $\operatorname{GL}_2$–$\operatorname{SL}_2$	$D$, $B'$ and $S$, $L$, $M$, $N$, $R^2$	Only $(L(D,1), N)$ and $(L(B',1), N)$ are proper
Both inside $$\operatorname{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2$$	$R^2$, $S$, $L$, $M$, $N$, $\operatorname{SL}_2(\mathbb{R})$	e.g. $\operatorname{SL}_2(\mathbb{R}) \bowtie R^2$, $R^2 \bowtie N$, $S \bowtie N$, $M \bowtie \operatorname{SL}_2$, etc.
Affine–nilpotent mixtures	$L(L)=Z,A(\alpha),U$ and $L(H)=R^2$, or vice versa	$(Z, R^2), (U, R^2), (A(\alpha), R^2)$ proper; others not.

Table 1: Properness among $1$-dimensional reductive subgroups

$L\,\backslash\,H$	$Z$	$A(\beta)$	$U$	$B(\beta)$
$Z$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
$A(\alpha)$	$\checkmark$	$|\alpha| \ne |\beta|$	$\checkmark$	$|\alpha| > |\beta|$
$U$	$\checkmark$	$\checkmark$	(never)	$\checkmark$
$B(\alpha)$	$\checkmark$	$|\alpha| < |\beta|$	$\checkmark$	(never)

Homogeneous spaces $G/H$ arising in these cases correspond to geometric models such as $\mathbb{R}^2$, $\mathbb{R} \times S^1$, the light cone, the cylinder, or Möbius strip, depending on $H$.

5. Reduction Lemmas and Structural Results

The classification relies on several major reduction principles:

• Proposition 2.1: For closed groups, $L \bowtie H$ if and only if $L$ acts properly on $G/H$.
• Theorem 7.3 (“sld–ed”): For $L$ in $\operatorname{SL}_n \ltimes \mathbb{R}^n$ but $H$ not, $L \bowtie H$ in $\operatorname{GL}_n \ltimes \mathbb{R}^n$ if and only if $$L \bowtie (H \cap \operatorname{SL}_n \ltimes \mathbb{R}^n)$$ in $\operatorname{SL}_n \ltimes \mathbb{R}^n$.
• Theorem 7.4 (RCI-criterion): If $(L,\mathbb{R}^n)$ is (CI) and $L(L) \bowtie L(H)$ in $\operatorname{GL}_n$, then $L \bowtie H$ in $\operatorname{Aff}(\mathbb{R}^n)$.
• The conjugacy classes of connected subalgebras of $\mathfrak{gl}_2 \ltimes \mathbb{R}^2$ (Chapovskyi–Koval–Zhur) enable effective reduction to finitely many normal forms, facilitating the exhaustive analysis (Miyauchi, 17 Jan 2026).

6. Relationship to Kobayashi’s Classification and Prior Work

Kobayashi (1992) had previously classified all connected closed subgroups of $\operatorname{Aff}(\mathbb{R}^2)$ acting properly on translation space $\mathbb{R}^2$—in other words, only pairs $(L, \mathbb{R}^2)$ with proper action. The current classification encompasses all possible connected proper pairs $(L, H)$, regardless of whether $H$ is purely translational, by combining the reductive Cartan projection criterion, the reduction to $\operatorname{SL}_2$ cases, and a systematic treatment of affine-nilpotent and mixed types. This generalizes and strictly extends the earlier results (Miyauchi, 17 Jan 2026).

7. Geometric Interpretation and Cartan Projection

Properness conditions for 1-dimensional reductive subgroups can be visualized via Cartan projections. The image under $\mu$ of each canonical type (e.g., $Z,\,A(\alpha),\,U,\,B(\beta)$) falls into specific regions of the positive Weyl chamber $a^+ \cong \{(x, y)\,|\,x \geq y \}$. Relative positions in this chamber directly determine whether particular proper pairs can exist. For instance, $A(\alpha) \bowtie A(\beta)$ holds if and only if $|\alpha| \ne |\beta|$, while $B(\alpha) \bowtie B(\beta)$ requires $|\alpha| < |\beta|$. The geometric structure of these images, and their mutual position, encapsulates the combinatorics of properness in the reductive setting.

A representative figure (see (Miyauchi, 17 Jan 2026)) visually encodes these projections, allowing at-a-glance determination of whether two types can form a proper pair based on their positions and overlap in the positive Weyl chamber.

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References

Miyauchi, "Classification of connected proper pairs in the affine transformation group of $\mathbb{R}^2$" (Miyauchi, 17 Jan 2026) Kobayashi, "Proper action on a homogeneous space of reductive type," Math. Ann. 285 (1989); "Proper actions on a homogeneous space of a reductive group," 1996. Chapovskyi–Koval–Zhur, classification of subalgebras of $\mathfrak{gl}_2 \ltimes \mathbb{R}^2$, 2024