On quasiconformal deformations of the universal hyperbolic solenoid
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 differentials. The points in \(\mathcal T(H_\infty)\) are the homotopy classes of the quasiconformal maps from the fixed complex solenoid \(H_\infty\) onto the variable solenoids \(X\). This implies that \(\mathcal T(H_\infty)\) is isomorphic to the set of equivalence classes of Beltrami coefficients on \(H_\infty\), where two Beltrami coefficients are equivalent if the corresponding quasiconformal maps are homotopic. This latter description of \(\mathcal T(H_\infty)\) is more convenient when considering the infinitesimal structure of \(\mathcal T(H_\infty)\). The Teichmüller distance from \([0]\in \mathcal T(H_\infty)\) to \([\mu]\in \mathcal T(H_\infty)\) is \(\frac{1}{2}\log\frac{1+k_0(\mu)}{1-k_0(\mu)}\) where \(k_0(\mu)=\inf_{\mu_1} \|\mu_1\|_\infty\) with the infimum over all Beltrami coefficients \(\mu_1\) Teichmüller equivalent to \(\mu\). We say that a Beltrami coefficient \(\mu\) is Teichmüller extremal if \(\|\mu\|_\infty=k_0(\mu)\). Note that extremal maps have Teichmüller extremal Beltrami coefficients and Teichmüller extremal Beltrami coefficients correspond to extremal maps. Therefore, to find the Teichmüller distance it is enough to find the Teichmüller extremal Beltrami coefficient. A Beltrami differential \(\mu\) is infinitesimally extremal if \(k_1(\mu)=\| \mu\|_\infty\) where \(k_1(\mu) = \inf_{\mu_1} \|\mu_1\|_\infty\) with the infimum over all \(\mu_1\) such that \(\mu-\mu_1\) is infinitesimally trivial. It is shown that a Beltrami coefficient \(\mu\) on the solenoid \(X\) is Teichmüller extremal if and only if it is infinitesimally extremal. As a consequence the author obtains a Teichmüller contraction principle: There exist constants \(C_1, C_2 > 0\) such that \[ C_1(\|\mu\|_\infty-k_0(\mu))\leq \|\mu\|_\infty -k_1(\mu) \leq C_2(\|\mu\|_\infty-k_0(\mu)) \] for any smooth Beltrami coefficient \(\mu\) on the solenoid \(X\) with \(\|\mu\|_\infty \leq k\), where \(0 < k < 1\) is fixed. In other words, the distance of an arbitrary Beltrami coefficient \(\mu\) from the extremal value \(k_0(\mu)\) of its Teichmüller class is comparable to its distance from the extremal value \(k_1(\mu)\) of its infinitesimal class. The norm \(k_1(\mu)\) of the infinitesimal class of a smooth Beltrami differential \(\mu\) on a solenoid \(X\) is equal to \(\sup_{\| \varphi\|=1}\text{Re}\int_X\mu\varphi\,dm\), where a tangent vector at \(X\in\mathcal T(H_\infty)\) is represented by the Beltrami differential \(\mu\) and the supremum is over all holomorphic quadratic differentials \(\varphi\) on \(X\) with norm 1. An analogue of the classical Bers embedding gives a complex Banach manifold structure on \(\mathcal T(H_\infty)\). The Kobayashi metric on \(\mathcal T(H_\infty)\) is defined using the complex structure on \(\mathcal T(H_\infty)\). It is equal to the Teichmüller metric as in the Riemann surface case.
holomorphic quadratic dif\-fe\-ren\-ti\-als
scientific article; zbMATH DE number 5351764
scientific article