FILLING LOOPS AT INFINITY IN THE MAPPING CLASS GROUP

Abstract. We study the Dehn function at infinity in the mapping class group, finding a polynomial upper bound of degree four. This is the same upper bound that holds for arbitrary right-angled Artin groups. Dehn functions quantify simple connectivity. That is, in a simply-connected space, every closed curve is the boundary of some disk; the Dehn function measures the area required to fill the curves of a given length. The growth of the Dehn function is invariant under quasi-isometry, so one can define the Dehn function not just for spaces, but also for groups. The Dehn function is not the only group invariant based on a filling problem; for example, one can also define the Dehn function at infinity, which is a quasi-isometry invariant that measures the difficulty of filling closed curves with disks that avoid a large ball. The Dehn function at infinity is a special case (k = 1) of the higher divergence functions Divk that were defined for groups in [1] and serve to quantify the connectivity at infinity. In that paper we survey some results using the growth rates of Divk to detect geometric features of groups and spaces