ar X iv : m at h / 05 12 16 2 v 2 [ m at h . G R ] 1 8 M ay 2 00 6 FREE AND FRAGMENTING FILLING LENGTH

A. The filling length of an edge-circuit η in the Cayley 2-complex of a finitely presented group is the least integer L such that there is a combinatorial null-homotopy of η down to a basepoint through loops of length at most L. We introduce similar notions in which the null-homotopy is not required to fix a basepoint, and in which the contracting loop is allowed to bifurcate. We exhibit groups in which the resulting filling invariants exhibit dramatically different behaviour to the standard notion of filling length. We also define the corresponding filling invariants for Riemannian manifolds and translate our results to this setting. 1. I Consider a vertical cylinder C ⊆ R 3 of height h whose base has diameter d ≪ h. Let S be the surface formed by the curved portion of C and the disc capping off its top. Topologi-cally, S is a closed 2-disc. The loop ∂S can be homotoped in S to a constant loop through loops of length at most πd by lifting it up the cylinder and then contracting it across the top of C. However, if we insist on keeping a basepoint on ∂S fixed in the course of the null-homotopy then we will encounter far longer loops, some of length at least 2h. In this article we will bring to light similar contrasts between basepoint-fixed and basepoint-free null-homotopies for loops in the Cayley 2-complex Cay 2 (P) of a finite presentation P of a group Γ. Words w that represent 1 in Γ (null-homotopic words) correspond to edge-circuits η w in Cay 2 (P). The filling length FL(w) of w was defined by Gromov [13] and in a combinatorial context is the minimal length L such that there is a basepoint-preserving combinatorial null-homotopy of η w through loops of length at most L. (A closely related notion called LNCH was considered by Gersten in [9].) We define FFL(w), the free filling length of w, likewise but without holding a basepoint fixed, and FFFL(w), the fragmenting free filling length of w, by also allowing the contracting loops to bifurcate. Detailed definitions are in Section 2. We construct a finite presentation in which FL(w) and FFL(w) differ dramatically for an infinite sequence of null-homotopic words of increasing length. [Our conventions are [a, b] = a −1 b −1 ab, a b = b −1 ab and a −b = b −1 a …