Inner radii of Teichmüller spaces of finitely generated Fuchsian groups (Q1190663)

Let \(T:=T(\Gamma)\) denote the Bers embedding of the Teichmüller space of the Fuchsian group \(\Gamma\) acting in the lower half plane \(L\). \(T\) is an open neighborhood of zero in a finite dimensional complex Banach space. The inradius \(i(\Gamma)\) of \(T\) is the radius of a maximal open ball \(B\subset T\) which is centered at 0. A classical result of Ahlfors and Weill says that \(i(\Gamma)\geq 2\). Nakanishi showed that the inequality is strict. Let \(I:=I(\Gamma)\) denote the infimum of \(i(\Gamma')\) where the infimum is taken over all Fuchsian groups \(\Gamma'\) acting in \(L\) which are quasiconformal deformations of \(\Gamma\). The author shows that \(I(\Gamma)=2\) if \(\Gamma\) is finitely generated of the second kind or is finitely generated with a Teichmüller space of positive dimension. A forthcoming result of Nakanishi and Velling that implies both results is mentioned.