Confirming Brennan’s Conjecture Numerically on a Counterexample to Thurston’s <i>K</i>  = 2 Conjecture (Q6613273)

Let \(\mathbb{H}^3\) be the 3-dimensional hyperbolic space. Let \(\Omega\) be a simply connected proper subdomain of \(\mathbb{C}\), and let \(\tilde{\Omega}\) be the union of all hyperbolic half-spaces \(H\) such that \(H \cap \partial \mathbb{H}^3 \subseteq \Omega\). The dome of \(\Omega\) is defined as \(\operatorname{Dome}(\Omega)=\partial \tilde{\Omega} \cap \mathbb{H}^3\). The sets \(\Omega\) and \(\operatorname{Dome}(\Omega)\) share the common boundary \(\partial \Omega\). Let \(\operatorname{M\ddot{o}b}(\Omega)\) be the set of all Möbius transformations that preserve \(\Omega\). Each such transformation extends to an isometry of \(\mathbb{H}^3\) preserving \(\operatorname{Dome}(\Omega)\). Let us define the following sets: \N\begin{align*}\N\mathcal{F} & =\{f: \Omega \rightarrow \operatorname{Dome}(\Omega): f(x)=x\ \text{for all}\ x \in \partial \Omega\ \text{and}\ f\ \text{is continuous}\}, \\\N\mathcal{F}_{\mathrm{eq}} & =\{f \in \mathcal{F}: f \circ \gamma=\gamma \circ f\ \text{for all}\ \gamma \in \operatorname{M\ddot{o}b}(\Omega)\}, \\\NK(\Omega) & = \inf \{K>0 : \text{there exists a } K\text{-quasiconformal map } f \in \mathcal{F}\},\\\NK_{eq}(\Omega) & = \inf \left\{K>0 : \text{there exists a}\ K\text{-quasiconformal map } f \in \mathcal{F}_{\text{eq}}\right\}.\N\end{align*}\NIt is known that \(\sup _{\Omega} K(\Omega)<\infty\) and \(\sup _{\Omega} K_{\mathrm{eq}}(\Omega)<\infty\). Thurston conjectured the following result (known as Thurston's \(K=2\) conjecture): For any simply connected proper subdomain \(\Omega \subset \mathbb{C}\), we have \(\sup _{\Omega} K(\Omega)=\) \(\sup _{\Omega} K_{\mathrm{eq}}(\Omega)=2\). Some counterexamples have been constructed by several authors to this conjecture.\N\NGiven a simply connected proper subdomain \(\Omega \subset \mathbb{C}\), there exists a conformal map \(F : \Omega \rightarrow \mathbb{D}\). Brennan conjectured about the growth of \(F'\) near \(\delta\Omega\) that \(F' \in L^p(\Omega)\ \text{for all}\ \frac{4}{3} < p < 4\). The aim of the paper under review is to investigate if Brennan's conjecture holds for a particular counterexample to Thurston's \(K\) = 2 conjecture. The author denotes by \(\Omega\) the counterexample given by \textit{Y. Komori} and \textit{C. A. Matthews} in [Conform. Geom. Dyn. 10, 184--196 (2006; Zbl 1116.30031)], where it was shown that \(K(\Omega)=K_{eq}(\Omega) > 2\). In the main result in the paper the author proves that Brennan's conjecture holds for \(\Omega\). More precisely: Let \(F: \Omega \rightarrow \mathbb{D}\) be a conformal map and set \(p_{\star}=\sup \left\{p>0: F^{\prime} \in L^p(\Omega)\right\}\). It is shown numerically that \(5.52<p_{\star}<5.54\). In particular, \(p_{\star}>4\) and so Brennan's conjecture holds for \(\Omega\).\N\NThe domain \(\Omega\) is a connected component of the domain of discontinuity of an explicit Kleinian once-punctured torus group. The author points out that his results strongly suggest that Brennan's conjecture holds for all domains constructed in this way and for all quasidisks invariant under a group of Möbius transformations, such that the quotient is a once-punctured torus.