## Abstract

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; we show it to be a group invariant and we give upper bounds on the gallery length of combable groups. We use gallery length to give a new proof of the Double Exponential Theorem. Also we give geometric inequalities relating gallery length to the space-complexity filling function known as filling length.

Translated title of the contribution | The gallery length filling function and a geometric inequality for filling length |
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Original language | English |

Pages (from-to) | 601 - 623 |

Number of pages | 23 |

Journal | Proceedings of the London Mathematical Society |

Volume | 92(3) |

DOIs | |

Publication status | Published - May 2006 |