Free and fragmenting filling length

Martin Bridson and Tim Riley

Preprint, December 2005. To appear in Journal Algebra

The filling length of an edge-circuit \eta in the Cayley 2-complex of a finite presentation of a group is the minimal integer length 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 base point, and in which the contracting loop is allowed to bifurcate. We exhibit a group 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.

19 pages, 9 figures.

[Pictures by Riley]