A closed ball compactification of a maximal component via cores of trees

TL;DR

The paper extends Thurston-type compactifications to the maximal component $Max(S)$ of the character variety for $G=PSL(2,\mathbb{R})\times PSL(2,\mathbb{R})$, showing that the natural compactification $\mathfrak B=Max(S)\cup \mathbb{P}Core(\mathcal{T},\mathcal{T})$ is a closed ball of dimension $12g-12$ with boundary described by $(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed structures. It connects degenerations of maximal representations to cores of product of $\mathbb{R}$-trees dual to measured laminations, via minimal Lagrangians in $\mathbb{H}^{2}\times\mathbb{H}^{2}$ and harmonic map parameterizations, and shows the mapping class group acts continuously on the compactification. The boundary objects are interpreted as mixed structures that dualize to subcomplexes in Euclidean buildings, unifying laminations, half-translation surfaces, and length-spectrum limits. The authors frame a broader conjecture for rank-2 groups and introduce Weyl-chamber valued laminations to capture vector-valued degeneration data, setting a path toward analogous compactifications for other higher Teichmüller components.

Abstract

We show that, in the character variety of surface group representations into the Lie group $\mathrm{PSL}(2,\mathbb{R}) \times \mathrm{PSL}(2,\mathbb{R})$, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as $(\overline{A_{1} \times A_{1}},2)$-valued mixed structures.

Theorems & Definitions (41)

• Theorem A
• Proposition 1.1
• Theorem B
• Theorem C
• Definition 3.1
• Proposition 3.2: Guirardel
• Lemma 3.3
• proof
• Lemma 3.4
• proof