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  or  for an English translation; see also  and ) 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  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.
Pablo Ares Gastesi 2005-08-31