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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.

MSC:
 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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