A geometric property of Bers’ embedding of the Teichmüller space. (English) Zbl 0458.30031

Abikoff, William

Riemann surfaces and related topics: Proc. 1978 Study Brook Conf., Ann. Math. Stud. 97, 3-5 (1981).

Let \( i: T(G) \rightarrow B \) be the Bers embedding of the Teichmuller space \( T(G) \) of a Fuchsian group \( G \) (of the first kind) into the space \( B \) of bounded quadratic differentials for \( G \); every point \( \varphi \) on the boundary of \( i(T(G)) \) in \( B \) corresponds to a Kleinian group \( G_{\varphi} \), a so-called b-group. Theorem. If \( \varphi \in \partial i(T(G)) \) and \( m(\varphi)=0 \) then \( \varphi \in \partial \operatorname{Ext}(i(\mathrm{~T}(\mathrm{G}))) \); here \( \mathrm{m}(\varphi) \) denotes the area of the limit set of the group \( G_{\varphi} \). At the above conference Thurston announced a proof that \( m(\varphi)=0 \) for all boundary groups of the Teichmiller space.

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MSC:

30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F35	Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30F10	Compact Riemann surfaces and uniformization

Keywords:

Teichmüller space of Fuchsian group; quadratic differentials

Citations:

Zbl 0447.00006