Conjugacy limits of certain subgroups in $\SL(2,\mathbb{R})\ltimes\mathbb{R}^2$

TL;DR

This paper studies conjugacy limits of several closed subgroup families inside $G=\operatorname{SL}(2,\mathbb{R})\ltimes \mathbb{R}^2$ using the Chabauty topology on the space of closed subgroups. By analyzing orbit closures under conjugation, the authors classify limit points for Levi subgroups, maximal compact subgroups, maximal diagonalizable subgroups, and the unipotent and Borel subgroups, revealing a rich boundary structure consisting of both familiar conjugates and new degenerate subgroups such as $N^+\ltimes \mathbb{R}^2$, $V_c$, $\tilde{N}^+$, and Heisenberg-type $V_{(a,b,c)}$. The results provide a detailed description of degenerations in a non-semisimple setting and connect to geometric transition concepts via subgroup limits. Overall, the work extends understanding of orbit closures in the Chabauty topology and informs potential applications to invariant random subgroups and geometric transitions.

Abstract

We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 6
• Remark 1
• proof
• Lemma 7
• proof