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The gallery length filling function and a geometric inequality for filling length

Steve M. Gersten and Tim R. Riley

Proc. London Math. Soc., 92(3), pages 601–623, May 2006

We exploit duality considerations in the study of singular combinatorial 2-discs ("diagrams") and are led to the following innovations concerning the geometry of the word problem for finite presentations of groups. We define a filling function called "gallery length" that measures the diameter of the 1-skeleton of the dual of diagrams, and we determine some of its properties. We use it to give a new and elementary proof of the Double Exponential Theorem. Also we give a geometric inequality for the space-complexity filling function known as filling length.

This article is the first in an intended series of three; the second is available here; a proof of the conjectures was intended to be the subject of the third. They were resolved negatively in November 2005 — details are here.