Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 577-599

TEICHMÜLLER THEORY OF THE UNIVERSAL HYPERBOLIC LAMINATION

Juan Manuel Burgos and Alberto Verjovsky

Centro de Investigación y de Estudios Avanzados, Departamento de Matemáticas
Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco
C.P. 07360 Ciudad de México, México; burgos 'at' math.cinvestav.mx

Universidad Nacional Autónoma de México, Instituto de Matemáticas – Unidad Cuernavaca
Av. Universidad S/N, C.P. 62210 Cuernavaca, Morelos, México; alberto 'at' matcuer.unam.mx

Abstract. We construct an Ahlfors–Bers complex analytic model for the Teichmüller space of the universal hyperbolic lamination (also known as Sullivan's Teichmüller space) and the renormalized Weil–Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivan's Teichmüller space is Kähler isometric biholomophic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichmüller space of this surface; i.e. we find natural Kähler coordinates for the Sullivan's Teichmüller space. This is the main result. As a corollary we show the expected fact that the Nag–Verjovsky embedding is transversal to the Sullivan's Teichmüller space contained in the universal one.

2010 Mathematics Subject Classification: Primary 32G05, 32G15, 57R30; Secondary 22C05, 32G81.

Key words: Teichmüller space, hyperbolic lamination, solenoid, Riemann surface.

Reference to this article: J. M. Burgos and A. Verjovsky: Teichmüller theory of the universal hyperbolic lamination. Ann. Acad. Sci. Fenn. Math. 45 (2020), 577-599.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4514