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On quasiconformal invariance of convergence and divergence types for Fuchsian groups. (English) Zbl 1189.30081

Let \(\Gamma\) be a Fuchsian group acting on the unit disc \(\mathbb{B}^2\), and denote by \[ P_s(\Gamma)= \sum_{\gamma\in\Gamma}\exp\Big(- s\rho(0,\gamma(0))\Big) \] the Poincaré series for \(\Gamma\), where \(\rho\) denotes the hyperbolic distance on \(\mathbb{B}^2\). The critical exponent \(\delta(\Gamma)\) of \(\Gamma\) is defined to be the abscissa of convergence of \(P_s(\Gamma)\), and \(\Gamma\) is said to be of convergence type if \(P_s(\Gamma)\) converges for \(s=\delta(\Gamma)\), and of divergence type otherwise. If \(\delta(\Gamma)= 1\), then \(\Gamma\) and all its quasiconformal conjugates are known to be either all of convergence type or all of divergence type. This result, however, does not extend to groups with \(\delta(\Gamma)< 1\).
The author’s main theorem states: There exists a Fuchsian group \(\Gamma\) of convergence type and a quasiconformal automorphism \(f\) of \(\mathbb{B}^2\) such that the Fuchsian group \(f^{-1}\Gamma f\) is of divergence type. To prove this result, the author first deduces a characterization of convergence and divergence types by considering certain modified Poincaré series which come up also in the construction of the Patterson measure. There are various intermediate results of independent interest.

MSC:

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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References:

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