Homological and Homotopical Higher-order Filling Functions

We construct groups in which FV (n) ≁ δ(n). This construction also leads to groups Gk, k ≥ 3 for which δ (n) is not subrecursive. The Dehn function of a group provides a measure of the complexity of the group’s word problem by measuring the difficulty of filling loops in a corresponding complex. A natural generalization is to consider the difficulty of filling higher-dimensional manifolds or cycles, and there are several ways to do so, varying in the nature of the filling and the boundary. One can consider, for example, the volume necessary to fill a k-sphere with a ball (δ), to fill ∂M with M (δ ), or to fill a (k − 1)-cycle by a k-chain (FV ). In some cases, these functions are equivalent; for example, the methods used in [11] work for all these definitions. Along these lines, Brady et al. [3] showed that if ∂M is connected and dimM = k+1 ≥ 4 then δ (n) δ(n). In this note, we will show that this is not necessarily true if dimM = 3, and that there are groups where FV 3 is not equivalent to δ. We will also show that for k ≥ 2 there are groups where FV k is not subrecursive (i.e., FV k grows faster than any computable function) and for k ≥ 3, there are groups where δ is not subrecursive. We start by defining some filling functions. To define δ, we will take the approach of Brady et al. [3], which is equivalent to the definition of Alonso, Wang, and Pride [2] or of Bridson [5]. We recall their definition of an admissible map: Definition 1 (Admissible maps [3]). Let W be a compact k-manifold and X a CW-complex. An admissible map from W to X is a map f : W → X such that f−1(X(k) \X(k−1)) is a disjoint union of open k-dimensional balls in W , each mapped homeomorphically to a k-cell of X. We define the volume vol(f) of f as the number of these balls.