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Bers' and Thurston's compactifications of Teichmüller space

The relationship between the compactifications of 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 denote the Teichmüller space of a compact surface of
genus . Given a pair of points and in , there
is a quasi-Fuchsian group , with two invariant components,
and , such that
and
. Conversely, a pair in
determines a
quasi-Fuchsian group with an isomorphism to the fundamental group of
. Fixing a element in the first factor of
we get an
embedding of Teichmüller space into the set
of conjugacy classes
of representations of into
. The
space
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
)
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 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 .

Pablo Ares Gastesi
2005-08-31