Free and fragmenting filling length

Abstract

The filling length of an edge-circuit \eta in the Cayley 2-complex of a finite presentation of a group is the least integer L such that there is a combinatorial null-homotopy of \eta down to a base point 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.