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The Gallery Length Filling Function and a Geometric Inequality for Filling Length
Author(s) -
Gersten S. M.,
Riley T. R.
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1017/s0024611505015649
Subject(s) - mathematics , invariant (physics) , function (biology) , exponential function , combinatorics , upper and lower bounds , geometry , mathematical analysis , evolutionary biology , mathematical physics , biology
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. 2000 Mathematics Subject Classification 20F05 (primary), 20F06, 57M05, 57M20 (secondary).