Modular groups – geometry and physics. (English) Zbl 0768.32015
Discrete groups and geometry, Proc. Conf., Birmingham/UK 1991, Lond. Math. Soc. Lect. Note Ser. 173, 94-103 (1992).
[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).
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).
Reviewer: W.Abikoff (Storrs)
MSC:
32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
30F60 | Teichmüller theory for Riemann surfaces |
30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
11F06 | Structure of modular groups and generalizations; arithmetic groups |