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The double exponential theorem for isodiametric and isoperimetric functions. (English) Zbl 0728.20029
An isoperimetric function for a finite presentation bounds the combinatorial area (that is, number of 2-cells) in minimal area van Kampen diagrams for relations while an isodiametric function bounds the minimal diameter of van Kampen diagrams for relations. A new proof is given for a result first proved by D. E. Cohen, that if \(f\colon\mathbb{N}\to\mathbb{N}\) is an isodiametric function for the finite presentation \(\mathcal P\), then there are constants \(a,b>1\) such that \(n\mapsto a^{b^{f(n)+n}}\) is an isoperimetric function for \(\mathcal P\). The method of proof involves an analysis of Stalling’s folding algorithm for finding a free basis for a finitely generated subgroup of a free group and relates such folds to Whitehead automorphisms of free groups. Crude bounds are also given for the word metric on \(\operatorname{Aut}(F)\), where \(F\) is a finitely generated free group.

20F06 Cancellation theory of groups; application of van Kampen diagrams
20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
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