Augmented Teichmüller spaces

William Abikoff
1976 Bulletin of the American Mathematical Society  
The augmented Teichmiiller space T, of a finitely generated Fuchsian group G of the first kind or a conformally finite Riemann surface S with signature, consists of the usual Teichmiiller space T together with the regular Z?-groups on its boundary. The structure of the regular Z?-groups has been studied in [2] (see also Marden [5] and Maskit [6] ). The usual topology on T given by the Bers embedding of T in the space of bounded quadratic differentials has a natural extension to T. The extension
more » ... to T. The extension corresponds to horocycles at the regular Z?-groups. It is discussed in §2. Some of the properties of T with this topology are listed below. Detailed proofs will appear elsewhere. A related study is being conducted by Earle and Marden. 1. Properties of T. THEOREM 1. Each element g of the Teichmiiller modular group, Mod, has a continuous extension to an automorphism of T. The proof of Theorem 1 follows from explicit construction of quasiconformal mappings realizing twist maps and transpositions. THEOREM 2. The augmented Riemann space R = T/Mod is a compact normal complex space. It is the unique compactification of R = T/Mod in the sense of Carton. The proof utilizes a correspondence between congruence classes of regular ^-groups and flags of subgroups of Mod. The uniqueness of the compactification together with results due to Bers [3] immediately yield THEOREM 3. R is a projective algebraic variety. By studying divergent sequences in T, we may prove the following conjecture of Ehrenpreis [4].
doi:10.1090/s0002-9904-1976-14049-9 fatcat:dmsi4gctkbh5xmgqmb4eci27fq