Fenchel-Nielsen coordinates for asymptotically conformal deformations (Q2790814)

For surfaces of infinite type various different Teichmüller spaces can be defined. The choice of a basepoint \(X\) allows to define, for instance, the length spectrum Teichmüller space \(T_{ls}(X)\), the quasiconformal Teichmüller space \(T_{qc}(X)\), and the ``little'' Teichmüller space \(T_0(X)\) consisting, respectively, of surfaces \(Y\) related to \(X\) by a homeomorphism with bounded stretch factor (\(T_{ls}(X)\)), by a quasiconformal map (\(T_{qc}(X)\)), and by an asymptotically conformal, quasiconformal map (\(T_0(X)\)).NEWLINENEWLINEThe choice of a bounded length pair of pants decomposition \(\mathcal P\) on \(X\) allows to define Fenchel-Nielsen coordinates (with values in \(\ell^\infty(\mathcal P)\times \ell^\infty(\mathcal P)\)). These give a locally bi-Lipschitz parametrization of \(T_{qc}(X)\) equipped with the Teichmüller metric, but are not surjective on the other spaces: the image of \(T_{qc}(X)\) was described by \textit{D. Alessandrini} et al. [Ann. Acad. Sci. Fenn., Math. 36, No. 2, 621--659 (2011; Zbl 1235.32010)].NEWLINENEWLINEIn this short note, the author describes the image of the little Teichmüller space \(T_0(X)\) and of its closure, \(\overline{T_0(X)}\), in the length spectrum metric under the Fenchel-Nielsen coordinates. This last result allows the author to show that the quotient \(T_{qc}/\overline{T_0}\), the so-called \textit{asymptotic Teichmüller space}, is contractible in the Teichmüller metric, and the quotients \(T_{ls}/\overline T_{qc}\) and \(T_{ls}/\overline{T_{0}}\) are contractible in the length spectrum metric.NEWLINENEWLINEThe aforementioned results build on Wolpert's inequality for lengths under \(K\)-quasiconformal mapping. In the last section of the paper, the author shows that this inequality is in general not sharp by finding sequences of simple closed curves whose length goes to infinity, but such that a quasiconformal map changes their length by at most an additive constant.