Bers' and Thurston's compactifications of Teichmüller space

The relationship between the compactifications of $ T(S)$ by Bers and Thurston was studied by Kerckhoff and Thurston in [29]. To explain the results of that paper we need to introduce a little notation. Let $ T_g$ denote the Teichmüller space of a compact surface of genus $ g \geq 2$. Given a pair of points $ X$ and $ Y$ in $ T_g$, there is a quasi-Fuchsian group $ G$, with two invariant components, $ \Omega$ and $ \Omega'$, such that $ \Omega/G = X$ and $ \Omega'/G = Y$. Conversely, a pair in $ T_g \times T_g$ determines a quasi-Fuchsian group with an isomorphism to the fundamental group of $ S$. Fixing a element in the first factor of $ T_g \times T_g$ we get an embedding of Teichmüller space into the set $ V(\pi_1(S))$ of conjugacy classes of representations of $ \pi_1(S)$ into $ \mathrm{PSL}(2,\mathbb{C})$. The space $ V(\pi_1(S))$ has a topology of convergence on finite system of generators. This embedding is holomorphic, and it is equivalent to the Bers embedding; the image is called a Bers slice, and its compactification is the Bers compactification of Teichmüller space. Changes of base point (the fixed point in the first factor of $ T_g \times T_g$) induce biholomorphic mappings between Bers slices.

Theorem 3.9   There are Bers slices for which the change of base point mappings do not extend to homeomorphisms between their compactifications.

It is also shown in the same paper that Bers' and Thurston's compactifications of Teichmüller space are not equivalent; this is a consequence of the following theorem.

Theorem 3.10   For $ g = 2$ there is a Bers slice for which the action of the modular group does not extend continuously to the compactification.

The authors sketch a proof of this result for any $ g \geq 2$.

Pablo Ares Gastesi 2005-08-31