On quasiconformal deformations of the universal hyperbolic solenoid
type: Item
instance of: scholarly article
title: On quasiconformal deformations of the universal hyperbolic solenoid
zbMATH Open document ID: 1161.30031
author: Dragomir Šarić
published in: Journal d'Analyse Mathématique
publication date: 2008-10-16T00:00:00Z
review text: The solenoid \(H_\infty\) is the inverse limit space of the system of all finite, unbranched coverings of a closed surface of genus at least two. Since the coverings are unbranched, every point in the universal hyperbolic solenoid \(H_\infty\) has an open neighborhood homeomorphic to (2-disk)\(\times\)(Cantor set). The global topology of the so\-le\-no\-id \(H_\infty\) is more complicated. Each path component of \(H_\infty\) is non-compact, simply connected and dense in the solenoid \(H_\infty\). The path components of \(H_\infty\) are homeomorphic to the unit disk with a non-standard topology. The author studies the space of deformations of the complex structure on \(H_\infty\), namely the Teichmüller space \(\mathcal T(H_\infty)\) of the universal hyperbolic solenoid \(H_\infty\). The focus is on the Teichmüller metric and the natural complex structure of \(\mathcal T(H_\infty)\). A Beltrami coefficient \(\mu\) on a complex solenoid \(X\) is called Teichmüller trivial if the corresponding quasiconformal map \(f: X \to X\) is homotopic to the identity. We say that \(\mu\) is infinitesimally trivial if \(\int_X\mu\varphi\,dm = 0\) for all holomorphic quadratic differentials \(\varphi\) on \(X\), where \(m\) is the properly normalized transverse measure on \(X\). A version of the Reich-Strebel inequality for complex solenoids \(X\) is proved: Let \(\varphi\) be a holomorphic quadratic differential on a complex solenoid \(X\) and let \(\mu\) be a Teichmüller trivial Beltrami coefficient. Then \[ \|\varphi\|\leq\int_X\frac{\left|1+\mu\frac{\varphi}{|\varphi|}\right|^2}{1-|\mu|^2}|\varphi|\,dm. \] As a consequence it is proved that the Teichmüller type Beltrami coefficients determine unique geodesics in \(\mathcal T(H_\infty)\). It is shown that the tangent space to \(\mathcal T(H_\infty)\) at a marked solenoid \(X\) is isomorphic to the quotient of the space of smooth Beltrami differentials on \(X\) by the subspace of infinitesimally trivial smooth Beltrami differentia…
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 5351764
zbMATH Keywords: Teichmüller space
zbMATH Keywords: holomorphic quadratic dif\-fe\-ren\-ti\-als
zbMATH Keywords: transverse me\-a\-su\-re
zbMATH Keywords: universal covering space
zbMATH Keywords: commensurable Poincaré theta series
zbMATH Keywords: universal hyperbolic solenoid
zbMATH Keywords: Reich-Strebel inequality
zbMATH Keywords: Beltrami coefficients
zbMATH Keywords: extremal maps
zbMATH Keywords: Kobayashi metric
MaRDI profile type: Publication
full work available at URL: s11854-008-0038-0
OpenAlex ID: W2008617883
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DOI: 10.1007/S11854-008-0038-0
description: scientific article; zbMATH DE number 5351764