Rank one phenomena for mapping class groups

Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmüller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indeed, the groups Mod(Σg) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic. A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σg) is similar to the former or the latter. Some higher rank behavior of Mod(Σg) is indicated by the cusp structure of moduli space, by the fact that Mod(Σg) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σg) of Tits’s Theorem on automorphism groups of higher rank buildings. In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σg) with lattices in rank one groups: every infinite order element of Mod(Σg) has linear growth in the word metric, and Mod(Σg) is not bound∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. †Supported in part by the US-Israel BSF grant. ‡Supported in part by NSF grant DMS 9971596