Filling Loops at Infinity in the Mapping Class Group

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 Div 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 Div to detect geometric features of groups and spaces. The mapping class group of a surface has quadratic Dehn function because it is automatic, and an automatic structure provides a combing which can be used to shrink a curve to a point using no more area than is needed in a Euclidean space (see [6, 3]). In this note we study the Dehn function at infinity: if we impose the additional condition that the filling of a loop avoid a large ball, must its area be much worse than quadratic? In [1, Theorem 6.1] we addressed this question and its higher-dimensional analogs in the case of right-angled Artin groups (RAAGs), and we showed that loops can be filled at infinity using area at most polynomial of degree four. Here we show that the same result holds in mapping class groups of surfaces of genus g ≥ 5, contributing to the growing literature comparing mapping class groups to RAAGs. We use two key features of these mapping class groups: first, they have presentations with short relators (due to Gervais [5]), and second, all abelian subgroups are undistorted [4]. Date: April 22, 2012. This work was supported by a SQuaRE grant from the American Institute of Mathematics. The second author and the fourth author are partially supported by NSF grants DMS-0906962 and DMS-0906086, respectively. The fifth author would like to thank New York University for its hospitality during the preparation of this paper. We thank the referee for helpful comments.