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On Dehn functions of free products of groups. (English) Zbl 0942.20023
Associated to a finite presentation of a group \(G\) is a Dehn function \(f\), where \(f(n)\) is the largest minimum number of conjugates of relators needed in a product that represents (in the free group on the generators) a word of length at most \(n\) that is trivial as an element of \(G\). Dehn functions for different finite presentations of \(G\) are equivalent in the sense that there is a positive integer \(C\) such that \(f(n)\leq Cg(Cn)+C\) and \(g(n)\leq Cf(Cn)+C\) for all \(n\). It is conjectured that every Dehn function is equivalent to a subnegative function, i.e., a function such that \(f(m)+f(n)\leq f(m+n)\) for all natural numbers \(m\) and \(n\). Any function \(f\) defined on the natural numbers has a subnegative closure \(\overline f\), defined by \(\overline f(n)=\max\{f(n_1)+\cdots+f(n_r)\}\) taken over all \(r\geq 1\) and all choices of \(n_j\) such that \(n_1+\cdots+n_r=n\). It is the smallest subnegative function with \(\overline f(n)\geq f(n)\) for all \(n\).
The main result of the paper is that if \(G_1\) and \(G_2\) are finitely presented nontrivial groups, then the Dehn function of \(G_1*G_2\) is equivalent to its subnegative closure. Using an inequality of S. G. Brick [Trans. Am. Math. Soc. 335, No. 1, 369-384 (1993; Zbl 0892.57001)], this implies that if \(f_i\) are Dehn functions for \(G_i\) for \(i=1,2\), then the Dehn function \(f\) of \(G_1*G_2\) is equivalent to \(\max\{\overline{f_1},\overline{f_2}\}\). The proof relies on careful examination and manipulation of diagrams.

20F65 Geometric group theory
57M07 Topological methods in group theory
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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