[For the entire collection see Zbl 0746.00069.]
This elegantly written paper has two related themes. The setting is the Teichmüller space \(T_{g,n}\) or \(T_ g\) of compact surfaces of genus \(g>1\) with \(n\) or no points removed. \(\text{Mod}_{g,n}\) or \(\text{Mod}_ g\) is the corresponding Teichmüller modular group. A Teichmüller disk is a proper holomorphic embedding of \(\Delta=\{| t|<1\}\subset\mathbb{C}\) into \(T_ g\) as the deformation of some surface \(S\in T_{g,n}\) defined by differentials \(t\omega\) on \(S\). Here \(\omega\) is a fixed holomorphic quadratic differential on \(S\) with at worst simple poles at the punctures.
The first theme concerns finding Teichmüller disks \(D_ T\) stabilized by the subgroup \(\Gamma_ D\) of \(\text{Mod}_ g\) with the property that \(D_ T/\Gamma_ D\) is a conformally finite Riemann surface. The construction proceeds by first finding a surface \(S\in D_ T\) and a holomorphic quadratic differential \(q\) on \(S\) which defines \(D_ T\). With sufficient symmetry, the desired surfaces in \(M_{g,n}/\Gamma_{g,n}\) are found. Examples are given where \(S\) is a Hecke surface or a Fermat surface.
The second theme is to consider the complex \({\mathcal T}_{g,n}\) of all systems of homotopy classes of simple closed loops on \(S\in T_{g,n}\). \({\mathcal T}_{g,n}\) is a deformation retract of \(T_{g,n}\) and, to each chamber of \({\mathcal T}_{g,n}\), there is associated a presentation of \(\text{Mod}_{g,n}\) given by a surjection from a braid group.
The paper closes with a description of the relations between these themes and areas such as string theory, rational billiards and applications to conformal field theory and topological field theory (in dimension 3).