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Linear growth of infinite order mapping class group elements

Fix a set of generators of the mapping class group of (the group
is finitely generated; there are many proofs of this fact, already known
to Dehn [14] or [15] for an English translation;
see also [31] and [24]) say
. We assume that the identity is not in
and
that if is in
then so is . The length of an
element of , , is defined as the minimum integer
such that can be expressed as a product of elements of
(with the length of the identity equal to 0).

Combining this result with the work of L. Mosher [38]
one gets that every element of infinite order in has linear
growth (theorem 1.2 in the cited paper).

If we now consider the Teichmüller metric on , and fix a point
, we have an embedding of the modular group on Teichmüller space by
. It is known that if a group acts with certain
properties on a space then this mapping is a quasi-isometry (see
[8, proposition 8.19]). However we have that this is
not true in the case under consideration.

**Theorem 4.3**
*The word metric on the modular group and the metric induced
by inclusion as an orbit in with the Teichmüller metric are not
Lipschitz equivalent.*
The proof is based on the fact that given Dehn twist , there exists a
totally geodesic copy of the upper half plane (Teichmüller disc), invariant
under , that contains and on which acts as a parabolic
transformation (
on
).

Pablo Ares Gastesi
2005-08-31