Weyl chamber length compactification of the PSL(2,R) x PSL(2,R) maximal character variety

• Verfasst von: Burger, Marc [VerfasserIn]
• Verfasst von: Iozzi, Alessandra [VerfasserIn]
• Verfasst von: Parreau, Anne [VerfasserIn]
• Verfasst von: Pozzetti, Maria Beatrice [VerfasserIn]
• Titel: Weyl chamber length compactification of the PSL(2,R) x PSL(2,R) maximal character variety
• Verf.angabe: Marc Burger, Alessandra Iozzi, Anne Parreau and Maria Beatrice Pozzetti
• Jahr: 2025
• Umfang: 23 S.
• Fussnoten: Online veröffentlicht: 27. September 2024 ; Gesehen am 26.03.2025
• Titel Quelle: Enthalten in: The Glasgow mathematical journal
• Ort Quelle: Cambridge : Cambridge Univ. Press, 1952
• Jahr Quelle: 2025
• Band/Heft Quelle: 67(2025), 1, Seite 11-33
• ISSN Quelle: 1469-509X
• DOI: doi:10.1017/S0017089524000156
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 20E08
• Sach-SW: 20F69
• Sach-SW: 22E40
• Sach-SW: 32G15
• Sach-SW: compactifications of character varieties
• Sach-SW: maximal representations
• Sach-SW: products of trees
• Sach-SW: surface groups
• K10plus-PPN: 1920590919

Abstract

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group ΓΓ\Gamma of a closed hyperbolic surface ΣΣ\Sigma in PSL(2,R)nPSL(2,R)n\textrm{PSL}(2,{\mathbb{R}})^n. We identify the boundary with the sphere P((ML)n)P((ML)n){\mathbb{P}}(({\mathcal{ML}})^n), where MLML\mathcal{ML} is the space of measured geodesic laminations on ΣΣ\Sigma. In the case n=2n=2n=2, we give a geometric interpretation of the boundary as the space of homothety classes of R2R2{\mathbb{R}}^2-mixed structures on ΣΣ\Sigma. We associate to such a structure a dual tree-graded space endowed with an R2+R2+{\mathbb{R}}_+^2-valued metric, which we show to be universal with respect to actions on products of two RR\mathbb{R}-trees with the given length spectrum.