2012/10/12 by Hideki Miyachi, Miyachi, Hideki
Mathematics · #20F38 (Secondary) #30F60 #32G15 (Primary) 31B15 #51F99 #57M99 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Holomorphic and Operator Theory #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1210.3431
openalex publication_date 2012/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we give a framework for the study of the extremal length\ngeometry of Teichm "uller space after S. Kerckhoff, F. Gardiner and H. Masur.\nThere is a natural compactification using extremal length geometry introduced\nby Gardiner and Masur. The compactification is realized in a certain projective\nspace. We develop the extremal length geometry in the cone which is defined as\nthe inverse image of the compactification via the quotient mapping. The\ncompactification is identified with a subset of the cone by taking an\nappropriate lift. The cone contains canonically the space of measured\nfoliations in the boundary.\n We first extend the geometric intersection number on the space of measured\nfoliations to the cone, and observe that the restriction of the intersection\nnumber to Teichm "uller space is represented explicitly by the formula in terms\nof the Gromov product with respect to the Teichm "uller distance. From this\nobservation, we deduce that the Gromov product extends continuously to the\ncompactification.\n As an application, we obtain an alternative approach to\nEarle-Ivanov-Kra-Markovic-Royden's characterization of isometries. Namely, with\nsome few exceptions, the isometry group of Teichm "uller space with respect to\nthe Teichm "uller distance is canonically isomorphic to the extended mapping\nclass group. We also obtain a new realization of Teichm "uller space, a\nhyperboloid model of Teichm "uller space with respect to the Teichm "uller\ndistance.\n